Exercise · ZW-COMBINATIONS-WITH-REPETITION-010 · ★★★ Advanced
How many integer solutions does \(x_1+x_2+x_3+x_4=9\) have with each \(x_i\ge 1\)?
Choose one option.
Show answer
- B. 56
Explanation
With \(y_i=x_i-1\ge 0\) the sum becomes 5: \(\binom{5+4-1}{5}=\binom{8}{5}=56\). The 220 counts the nonnegative ones with sum 9.
Review the concept: Combinations with repetition
A combination with repetition of size \(k\) from \(n\) types of object is an unordered selection of \(k\) objects in which each type may be chosen several times.
\[\binom{n+k-1}{k}\]
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