To solve the problem of finding the number of positive integer solutions to the equation with 5 variables summing to a given number, we use the stars and bars method.

Key Insight:

For the equation (x_1 + x_2 + x_3 + x_4 + x_5 = N) where each (x_i) is a positive integer:

• Let (y_i = x_i - 1) (so (y_i \geq 0)).
• The equation becomes (y_1 + y_2 + ... + y_5 = N - 5).

The number of non-negative solutions to this is given by the combination formula:
[ \binom{(N-5) + 5 - 1}{5 - 1} = \binom{N-1}{4} ]

Assumption (common problem setup):

If the right-hand side (N = 10) (a typical value for such problems):
[ \binom{10-1}{4} = \binom{9}{4} = 126 ]

Answer: (\boxed{126})

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