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What is the coefficient of $x^2y^6$ in the expansion of $\left(\frac{3}{5}x-\frac{y}{2}\right)^8$? Express your answer as a common fraction

To find the coefficient of \( x^2y^6 \) in the expansion of \( \left(\frac{3}{5}x - \frac{y}{2}\right)^8 \), we will use the Binomial Theorem, which states that: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] In our case, we can identify \( a = \frac{3}{5}x \) and \( b = -\frac{y}{2} \), and \( n = 8 \). ### Step 1: Analyze the problem requirements We need to find the specific term in the expansion that contains \( x^2y^6 \). This means we need to determine the values of \( k \) and \( n-k \) such that: - The power of \( x \) is 2. - The power of \( y \) is 6. ### Step 2: List the steps to solve the problem 1. Identify the general term in the binomial expansion. 2. Set up equations based on the powers of \( x \) and \( y \). 3. Solve for \( k \) and \( n-k \). 4. Substitute \( k \) into the general term to find the coefficient. 5. Simplify the coefficient to express it as a common fraction. ### Step 3: Execute the solution process 1. The general term in the expansion is given by: \[ T_k = \binom{8}{k} \left(\frac{3}{5}x\right)^{8-k} \left(-\frac{y}{2}\right)^k \] 2. We need: - The power of \( x \) from \( \left(\frac{3}{5}x\right)^{8-k} \) is \( 8-k \). - The power of \( y \) from \( \left(-\frac{y}{2}\right)^k \) is \( k \). We want: \[ 8-k = 2 \quad \text{(for } x^2\text{)} \] \[ k = 6 \quad \text{(for } y^6\text{)} \] 3. Solving \( 8-k = 2 \) gives: \[ k = 6 \] 4. Now we substitute \( k = 6 \) into the general term: \[ T_6 = \binom{8}{6} \left(\frac{3}{5}x\right)^{8-6} \left(-\frac{y}{2}\right)^6 \] This simplifies to: \[ T_6 = \binom{8}{6} \left(\frac{3}{5}x\right)^{2} \left(-\frac{y}{2}\right)^{6} \] 5. Calculate \( \binom{8}{6} \): \[ \binom{8}{6} = \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28 \] 6. Now calculate \( \left(\frac{3}{5}\right)^2 \) and \( \left(-\frac{1}{2}\right)^6 \): \[ \left(\frac{3}{5}\right)^2 = \frac{9}{25} \] \[ \left(-\frac{1}{2}\right)^6 = \frac{1}{64} \] 7. Combine these to find the coefficient: \[ T_6 = 28 \cdot \frac{9}{25} \cdot \frac{1}{64} \] \[ = 28 \cdot \frac{9}{1600} \] \[ = \frac{252}{1600} \] 8. Simplify \( \frac{252}{1600} \): - The GCD of 252 and 1600 is 4. \[ \frac{252 \div 4}{1600 \div 4} = \frac{63}{400} \] ### Step 4: Provide the final answer The coefficient of \( x^2y^6 \) in the expansion of \( \left(\frac{3}{5}x - \frac{y}{2}\right)^8 \) is: \[ \boxed{\frac{63}{400}} \]

how many r in strawberry?

To determine how many times the letter "r" appears in the word "strawberry," we will follow a systematic approach. ### Step 1: Analyze the problem requirements The problem requires us to count the occurrences of the letter "r" in the word "strawberry." This is a straightforward counting problem where we need to identify each instance of the letter "r" in the given word. ### Step 2: List the steps to solve the problem 1. Write down the word "strawberry." 2. Identify and count each occurrence of the letter "r" in the word. 3. Provide the total count of the letter "r." ### Step 3: Execute the solution process 1. The word we are analyzing is "strawberry." 2. Now, let's break down the word and look for the letter "r": - The letters in "strawberry" are: s, t, r, a, w, b, e, r, r, y. 3. We can see that: - The first "r" appears in the third position. - The second "r" appears in the eighth position. - The third "r" appears in the ninth position. 4. Counting these occurrences, we find: - 1st "r" (3rd position) - 2nd "r" (8th position) - 3rd "r" (9th position) ### Step 4: Provide the final answer The total number of occurrences of the letter "r" in the word "strawberry" is: \[ \boxed{3} \]

how many e in 'The quick brown fox jumps over the lazy dog.'?

To determine how many times the letter "e" appears in the sentence "The quick brown fox jumps over the lazy dog," we will follow a systematic approach. ### Step 1: Analyze the problem requirements The problem requires us to count the occurrences of the letter "e" in the given sentence. This is a straightforward counting problem where we need to identify each instance of the letter "e," regardless of its case (uppercase or lowercase). ### Step 2: List the steps to solve the problem 1. Write down the sentence: "The quick brown fox jumps over the lazy dog." 2. Identify and count each occurrence of the letter "e" in the sentence. 3. Provide the total count of the letter "e." ### Step 3: Execute the solution process 1. The sentence we are analyzing is: "The quick brown fox jumps over the lazy dog." 2. Now, let's break down the sentence and look for the letter "e": - We can go through the sentence letter by letter or group by words. - The words in the sentence are: The, quick, brown, fox, jumps, over, the, lazy, dog. 3. Now, let's count the occurrences of "e": - In "The," there is **1** "e." - In "quick," there are **0** "e." - In "brown," there are **0** "e." - In "fox," there are **0** "e." - In "jumps," there are **0** "e." - In "over," there is **1** "e." - In "the," there is **1** "e." - In "lazy," there are **0** "e." - In "dog," there are **0** "e." 4. Now, we can sum the occurrences: - From "The": 1 - From "over": 1 - From "the": 1 - Total occurrences of "e": \(1 + 1 + 1 = 3\) ### Step 4: Provide the final answer The total number of occurrences of the letter "e" in the sentence "The quick brown fox jumps over the lazy dog" is: \[ \boxed{3} \]

how many t in 'She sells seashells by the seashore.'?

To determine how many times the letter "t" appears in the sentence "She sells seashells by the seashore," we will follow a systematic approach. ### Step 1: Analyze the problem requirements The problem requires us to count the occurrences of the letter "t" in the given sentence. This is a straightforward counting problem where we need to identify each instance of the letter "t," regardless of its case (uppercase or lowercase). ### Step 2: List the steps to solve the problem 1. Write down the sentence: "She sells seashells by the seashore." 2. Identify and count each occurrence of the letter "t" in the sentence. 3. Provide the total count of the letter "t." ### Step 3: Execute the solution process 1. The sentence we are analyzing is: "She sells seashells by the seashore." 2. Now, let's break down the sentence and look for the letter "t": - We can go through the sentence letter by letter or group by words. - The words in the sentence are: She, sells, seashells, by, the, seashore. 3. Now, let's count the occurrences of "t": - In "She," there are **0** "t." - In "sells," there are **0** "t." - In "seashells," there are **0** "t." - In "by," there are **0** "t." - In "the," there is **1** "t." - In "seashore," there are **0** "t." 4. Now, we can sum the occurrences: - From "She": 0 - From "sells": 0 - From "seashells": 0 - From "by": 0 - From "the": 1 - From "seashore": 0 - Total occurrences of "t": \(0 + 0 + 0 + 0 + 1 + 0 = 1\) ### Step 4: Provide the final answer The total number of occurrences of the letter "t" in the sentence "She sells seashells by the seashore" is: \[ \boxed{1} \]

how many a in 'Aardvarks are amazing animals.'?

To determine how many times the letter "a" appears in the sentence "Aardvarks are amazing animals," we will follow a systematic approach. ### Step 1: Analyze the problem requirements The problem requires us to count the occurrences of the letter "a" in the given sentence. This includes both uppercase "A" and lowercase "a." We need to ensure that we accurately identify each instance of the letter "a" regardless of its case. ### Step 2: List the steps to solve the problem 1. Write down the sentence: "Aardvarks are amazing animals." 2. Identify and count each occurrence of the letter "a" in the sentence. 3. Provide the total count of the letter "a." ### Step 3: Execute the solution process 1. The sentence we are analyzing is: "Aardvarks are amazing animals." 2. Now, let's break down the sentence and look for the letter "a": - We can go through the sentence letter by letter or group by words. - The words in the sentence are: Aardvarks, are, amazing, animals. 3. Now, let's count the occurrences of "a": - In "Aardvarks," there are **3** "a" (1 uppercase "A" and 2 lowercase "a"). - In "are," there is **1** "a." - In "amazing," there is **1** "a." - In "animals," there is **1** "a." 4. Now, we can sum the occurrences: - From "Aardvarks": 3 - From "are": 1 - From "amazing": 1 - From "animals": 1 - Total occurrences of "a": \(3 + 1 + 1 + 1 = 6\) ### Step 4: Provide the final answer The total number of occurrences of the letter "a" in the sentence "Aardvarks are amazing animals" is: \[ \boxed{6} \]

how many o in 'Once upon a time in a land far, far away.'?

To determine how many times the letter "o" appears in the sentence "Once upon a time in a land far, far away," we will follow a systematic approach. ### Step 1: Analyze the problem requirements The problem requires us to count the occurrences of the letter "o" in the given sentence. This includes both uppercase "O" and lowercase "o." We need to ensure that we accurately identify each instance of the letter "o" regardless of its case. ### Step 2: List the steps to solve the problem 1. Write down the sentence: "Once upon a time in a land far, far away." 2. Identify and count each occurrence of the letter "o" in the sentence. 3. Provide the total count of the letter "o." ### Step 3: Execute the solution process 1. The sentence we are analyzing is: "Once upon a time in a land far, far away." 2. Now, let's break down the sentence and look for the letter "o": - We can go through the sentence letter by letter or group by words. - The words in the sentence are: Once, upon, a, time, in, a, land, far, far, away. 3. Now, let's count the occurrences of "o": - In "Once," there is **1** "o." - In "upon," there is **1** "o." - In "a," there are **0** "o." - In "time," there are **0** "o." - In "in," there are **0** "o." - In "a," there are **0** "o." - In "land," there are **0** "o." - In "far," there are **0** "o." - In "far," there are **0** "o." - In "away," there are **0** "o." 4. Now, we can sum the occurrences: - From "Once": 1 - From "upon": 1 - From "a": 0 - From "time": 0 - From "in": 0 - From "a": 0 - From "land": 0 - From "far": 0 - From "far": 0 - From "away": 0 - Total occurrences of "o": \(1 + 1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 2\) ### Step 4: Provide the final answer The total number of occurrences of the letter "o" in the sentence "Once upon a time in a land far, far away" is: \[ \boxed{2} \]

how many i in 'The significance of this discovery is immense.'?

To determine how many times the letter "i" appears in the sentence "The significance of this discovery is immense," we will follow a systematic approach. ### Step 1: Analyze the problem requirements The problem requires us to count the occurrences of the letter "i" in the given sentence. This includes both uppercase "I" and lowercase "i." We need to ensure that we accurately identify each instance of the letter "i" regardless of its case. ### Step 2: List the steps to solve the problem 1. Write down the sentence: "The significance of this discovery is immense." 2. Identify and count each occurrence of the letter "i" in the sentence. 3. Provide the total count of the letter "i." ### Step 3: Execute the solution process 1. The sentence we are analyzing is: "The significance of this discovery is immense." 2. Now, let's break down the sentence and look for the letter "i": - We can go through the sentence letter by letter or group by words. - The words in the sentence are: The, significance, of, this, discovery, is, immense. 3. Now, let's count the occurrences of "i": - In "The," there are **0** "i." - In "significance," there are **3** "i." - In "of," there are **0** "i." - In "this," there is **1** "i." - In "discovery," there is **1** "i." - In "is," there is **1** "i." - In "immense," there are **0** "i." 4. Now, we can sum the occurrences: - From "The": 0 - From "significance": 3 - From "of": 0 - From "this": 1 - From "discovery": 1 - From "is": 1 - From "immense": 0 - Total occurrences of "i": \(0 + 3 + 0 + 1 + 1 + 1 + 0 = 6\) ### Step 4: Provide the final answer The total number of occurrences of the letter "i" in the sentence "The significance of this discovery is immense" is: \[ \boxed{6} \]

how many s in 'Success is not final, failure is not fatal.'?

To determine how many times the letter "s" appears in the sentence "Success is not final, failure is not fatal," we will follow a systematic approach. ### Step 1: Analyze the problem requirements The problem requires us to count the occurrences of the letter "s" in the given sentence. This includes both uppercase "S" and lowercase "s." We need to ensure that we accurately identify each instance of the letter "s" regardless of its case. ### Step 2: List the steps to solve the problem 1. Write down the sentence: "Success is not final, failure is not fatal." 2. Identify and count each occurrence of the letter "s" in the sentence. 3. Provide the total count of the letter "s." ### Step 3: Execute the solution process 1. The sentence we are analyzing is: "Success is not final, failure is not fatal." 2. Now, let's break down the sentence and look for the letter "s": - We can go through the sentence letter by letter or group by words. - The words in the sentence are: Success, is, not, final, failure, is, not, fatal. 3. Now, let's count the occurrences of "s": - In "Success," there are **2** "s." - In "is," there is **1** "s." - In "not," there are **0** "s." - In "final," there are **0** "s." - In "failure," there are **0** "s." - In "is," there is **1** "s." - In "not," there are **0** "s." - In "fatal," there are **0** "s." 4. Now, we can sum the occurrences: - From "Success": 2 - From "is": 1 - From "not": 0 - From "final": 0 - From "failure": 0 - From "is": 1 - From "not": 0 - From "fatal": 0 - Total occurrences of "s": \(2 + 1 + 0 + 0 + 0 + 1 + 0 + 0 = 4\) ### Step 4: Provide the final answer The total number of occurrences of the letter "s" in the sentence "Success is not final, failure is not fatal" is: \[ \boxed{4} \]

how many r in 'The rain in Spain falls mainly on the plain.'?

To determine how many times the letter "r" appears in the sentence "The rain in Spain falls mainly on the plain," we will follow a systematic approach. ### Step 1: Analyze the problem requirements The problem requires us to count the occurrences of the letter "r" in the given sentence. This includes both uppercase "R" and lowercase "r." We need to ensure that we accurately identify each instance of the letter "r" regardless of its case. ### Step 2: List the steps to solve the problem 1. Write down the sentence: "The rain in Spain falls mainly on the plain." 2. Identify and count each occurrence of the letter "r" in the sentence. 3. Provide the total count of the letter "r." ### Step 3: Execute the solution process 1. The sentence we are analyzing is: "The rain in Spain falls mainly on the plain." 2. Now, let's break down the sentence and look for the letter "r": - We can go through the sentence letter by letter or group by words. - The words in the sentence are: The, rain, in, Spain, falls, mainly, on, the, plain. 3. Now, let's count the occurrences of "r": - In "The," there are **0** "r." - In "rain," there is **1** "r." - In "in," there are **0** "r." - In "Spain," there is **1** "r." - In "falls," there are **0** "r." - In "mainly," there are **0** "r." - In "on," there are **0** "r." - In "the," there are **0** "r." - In "plain," there are **0** "r." 4. Now, we can sum the occurrences: - From "The": 0 - From "rain": 1 - From "in": 0 - From "Spain": 1 - From "falls": 0 - From "mainly": 0 - From "on": 0 - From "the": 0 - From "plain": 0 - Total occurrences of "r": \(0 + 1 + 0 + 1 + 0 + 0 + 0 + 0 + 0 = 2\) ### Step 4: Provide the final answer The total number of occurrences of the letter "r" in the sentence "The rain in Spain falls mainly on the plain" is: \[ \boxed{2} \]

how many n in 'Necessity is the mother of invention.'?

To determine how many times the letter "n" appears in the sentence "Necessity is the mother of invention," we will follow a systematic approach. ### Step 1: Analyze the problem requirements The problem requires us to count the occurrences of the letter "n" in the given sentence. This includes both uppercase "N" and lowercase "n." We need to ensure that we accurately identify each instance of the letter "n" regardless of its case. ### Step 2: List the steps to solve the problem 1. Write down the sentence: "Necessity is the mother of invention." 2. Identify and count each occurrence of the letter "n" in the sentence. 3. Provide the total count of the letter "n." ### Step 3: Execute the solution process 1. The sentence we are analyzing is: "Necessity is the mother of invention." 2. Now, let's break down the sentence and look for the letter "n": - We can go through the sentence letter by letter or group by words. - The words in the sentence are: Necessity, is, the, mother, of, invention. 3. Now, let's count the occurrences of "n": - In "Necessity," there is **1** "n." - In "is," there are **0** "n." - In "the," there are **0** "n." - In "mother," there is **1** "n." - In "of," there are **0** "n." - In "invention," there are **2** "n." 4. Now, we can sum the occurrences: - From "Necessity": 1 - From "is": 0 - From "the": 0 - From "mother": 1 - From "of": 0 - From "invention": 2 - Total occurrences of "n": \(1 + 0 + 0 + 1 + 0 + 2 = 4\) ### Step 4: Provide the final answer The total number of occurrences of the letter "n" in the sentence "Necessity is the mother of invention" is: \[ \boxed{4} \]

how many l in 'Lollipops are loved by little children.'?

To determine how many times the letter "l" appears in the sentence "Lollipops are loved by little children," we will follow a systematic approach. ### Step 1: Analyze the problem requirements The problem requires us to count the occurrences of the letter "l" in the given sentence. This includes both uppercase "L" and lowercase "l." We need to ensure that we accurately identify each instance of the letter "l" regardless of its case. ### Step 2: List the steps to solve the problem 1. Write down the sentence: "Lollipops are loved by little children." 2. Identify and count each occurrence of the letter "l" in the sentence. 3. Provide the total count of the letter "l." ### Step 3: Execute the solution process 1. The sentence we are analyzing is: "Lollipops are loved by little children." 2. Now, let's break down the sentence and look for the letter "l": - We can go through the sentence letter by letter or group by words. - The words in the sentence are: Lollipops, are, loved, by, little, children. 3. Now, let's count the occurrences of "l": - In "Lollipops," there are **2** "l" (1 uppercase and 1 lowercase). - In "are," there are **0** "l." - In "loved," there is **1** "l." - In "by," there are **0** "l." - In "little," there are **3** "l." - In "children," there is **1** "l." 4. Now, we can sum the occurrences: - From "Lollipops": 2 - From "are": 0 - From "loved": 1 - From "by": 0 - From "little": 3 - From "children": 1 - Total occurrences of "l": \(2 + 0 + 1 + 0 + 3 + 1 = 7\) ### Step 4: Provide the final answer The total number of occurrences of the letter "l" in the sentence "Lollipops are loved by little children" is: \[ \boxed{7} \]

how many c in 'Circumstances can change quickly.'?

To determine how many times the letter "c" appears in the sentence "Circumstances can change quickly," we will follow a systematic approach. ### Step 1: Analyze the problem requirements The problem requires us to count the occurrences of the letter "c" in the given sentence. This includes both uppercase "C" and lowercase "c." We need to ensure that we accurately identify each instance of the letter "c" regardless of its case. ### Step 2: List the steps to solve the problem 1. Write down the sentence: "Circumstances can change quickly." 2. Identify and count each occurrence of the letter "c" in the sentence. 3. Provide the total count of the letter "c." ### Step 3: Execute the solution process 1. The sentence we are analyzing is: "Circumstances can change quickly." 2. Now, let's break down the sentence and look for the letter "c": - We can go through the sentence letter by letter or group by words. - The words in the sentence are: Circumstances, can, change, quickly. 3. Now, let's count the occurrences of "c": - In "Circumstances," there are **2** "c" (1 uppercase and 1 lowercase). - In "can," there is **1** "c." - In "change," there is **1** "c." - In "quickly," there are **0** "c." 4. Now, we can sum the occurrences: - From "Circumstances": 2 - From "can": 1 - From "change": 1 - From "quickly": 0 - Total occurrences of "c": \(2 + 1 + 1 + 0 = 4\) ### Step 4: Provide the final answer The total number of occurrences of the letter "c" in the sentence "Circumstances can change quickly" is: \[ \boxed{4} \]