Question

How many solutions are there to the inequality, where, andare nonnegative integers? (Hint: Introduce an auxiliary variable x4 such that.)

Short answer

364 ways solutions are present in the inequality

Step-by-step solution

Step 1: Use the formula for integer

Formula for integer

\(C\left( {n + r - 1,r} \right) = \frac{{(n + r - 1)!}}{{n!r!}}\)

n is total number of auxiliary variables

r is constant

Step 2: Solution of number ways in equality equation

Let’s, applied n and r values of inequality in integer formula

and all the 3 integers are non-negative.

Add new auxiliary variable ofsuch thatand

is an integer so count total number of auxiliary variable is 4

Here,

\(\begin{array}{l}C\left( {n + r - 1,r} \right) = \frac{{(n + r - 1)!}}{{n!r!}}\\n = 4\\r = 11\\C\left( {n + r - 1,r} \right) = C\left( {4 + 11 - 1,11} \right)\\ = C\left( {14,11} \right)\\ = \frac{{14!}}{{11!4!}}\\ = 364ways\end{array}\)