# While solving an LPP (defined by n variables and m equations, m < n) through simplex method, basic solutions are determined by setting n – m variables equal to zero and solving m equations to obtain solution for remaining m variables, provided the resulting solutions are unique. This means that the maximum number of basic solutions is:

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1. $$\frac{{n!}}{{m!\left( {n - m} \right)!}}$$
2. $$\frac{{m!}}{{n!\left( {n - m} \right)!}}$$
3. $$\frac{{n!}}{{m!\left( {n + m} \right)!}}$$
4. $$\frac{{m!}}{{n!\left( {n + m} \right)!}}$$

Option 1 : $$\frac{{n!}}{{m!\left( {n - m} \right)!}}$$
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TNTRB 2017 ME Official Question Paper
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150 Questions 190 Marks 180 Mins

## Detailed Solution

Explanation:

• The standard LPP form includes "m" simultaneous linear equations in "n" variables. i.e. m < n.
• The "n" variables are divided into two sets:
• n-m variables, to which we assign zero values, and
• The remaining "m" variables, whose values are determined by solving the resulting "m" equations.
• If the m equations yield a unique solution, then the associated m variables are called basic variables, and the remaining n - m zero variables are referred to as non-basic variables.
• In this case, the resulting unique solution comprises a basic solution. If all the variables assume non-negative values, then the basic solution is feasible. otherwise, it is infeasible.
• By definition, the maximum number of possible basic solutions for 'm' equations in n unknowns is
• $$\left( {\begin{array}{*{20}{c}} n\\ m \end{array}} \right) = \frac{{n!}}{{m!\left( {m - n} \right)!}}$$