First slide

Combinations

Question

The number of ordered triplets of positive integers which are solutions of the equation x + y + z = 100 is

Moderate

A

5081
B

6005
C

4851
D

None of these

Solution

The number of triplets of positive integers which are solutions of x + y + z = 100
$\begin{array}{l} = Coefficient of x^{100} in {(x + x^{2} + x^{3} + \dots)}^{3} \\ = Coefficient of x^{100} in x^{3} (1 - x)^{- 3} \\ = Coefficient of x^{100} in x^{3} [1 + 3 x + 6 x^{2} + \dots + \frac{(n + 1) (n + 2)}{2} x^{n} + \dots] \\ = \frac{(97 + 1) (97 + 2)}{2} = 49 \times 99 = 4851 \end{array}$

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