Question

How many natural solution of equation x1+x2+x3+x4 = 25 such that x1 and x2 are even?

Answer 1

Answer:

given down.

Step-by-step explanation:

We can solve this problem in a case where ``natural number" is meant to include zero (see Is zero a natural number? Why or why not?) Otherwise, we can just say that x1=1+y1, etc., and solve the same problem with the sum on the right-hand side being smaller by 3.

There is a general way to find the number of solutions to x1+x2+...+xm=n.  Suppose that we have n balls and m−1 dividers (all indistinguishable).  Then there is a one-to-one mapping between solutions to the equation above and arrangements between balls and dividers (by taking x1 to be the number of balls to the left of the first divider, x2 the number of balls between the first and second, etc.)   The number of such arrangements is just the number of ways to choose n elements from a set of n+m−1, which is (n+m−1m−1).  In this case, it would be (132) or 78 (if we exclude zero, it would be (102) or 45).