First slide

Combinations

Question

The number of positive integral solutions of $15 < x_{1} + x_{2} + x_{3} \leq 20$ , is equal to

Moderate

A

785
B

685
C

1150
D

None of these

Solution

We have, $15 < x_{1} + x_{2} + x_{3} \leq 20$
$\Rightarrow x_{1} + x_{2} + x_{3} = 16 + r, r = 0, 1, 2, 3, 4$ .
Now, number of positive integral solutions of $x_{1} + x_{2} + x_{3} = 16 + r$ is ${}^{16 + r - 1}C_{3 - 1} = {}^{15 + r}C_{2}$
Thus, total number of solutions
$= \sum_{r = 0}^{4}^{15 + r} {C_{2}}^{= 15} C_{2} +^{16} C_{2} +^{17} C_{2} +^{18} C_{2} +^{19} {C_{2}}^{= 20} C_{3} -^{15} C_{3} = 685$

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