# Tutoring math, you might be asked counting problems. The tutor brings up one.

How many solutions can be found to x_{1} + x_{2} + x_{3} + x_{4} = 10, where x_{1}, x_{2}, x_{3}, and x_{4} are all whole numbers?

Imagine three pipes |||, and ten asterisks **********. Then 5, for example, appears as *****. The pipes separate the values of x_{1}, x_{2}, x_{3}, and x_{4}. Therefore, the solution x_{1}=3, x_{2} =0,x_{3}=5, x_{4}=2 shows as

***||*****|**

By the guidelines above, any solution to x_{1} + x_{2} + x_{3} + x_{4} = 10 can be shown as a sequence of 13 characters: three pipes and ten asterisks. The variability is which three positions the pipes occupy. 13C3 is how many ways the pipes can be distributed. Therefore, there are 13C3 solutions to x_{1} + x_{2} + x_{3} + x_{4} = 10, where x_{1}, x_{2}, x_{3}, and x_{4} are all whole numbers.

Source:

Ross, Sheldon. *A First Course in Probability*. New York: Macmillan Publishing Company, 1988.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

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