Question

Find the number of distinct integer 

solutions of the equation : x₁+x₂+ ......+x₅=24

x_i≥i and 1≤i≤5

Answer 1

Let $y_i=x_i-i$ for $1\leq i\leq5.$ We want to count solutions to

$$y_1+y_2+y_3+y_4+y_5=24-1-2-3-4-5=9$$

There are $C(9+ 5 - 1, 5 - 1)$ ways to place $5$ indistinguishable objects into $9$ distinguishable boxes

$$\dbinom{13}{4}=\dfrac{13!}{4!(13-4)!}=715$$

There are 715 solutions.