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Permutations and Combinations

If set A = r ...

Question

If set $A = {(r, s) | r < s; r, s \in W}$ , then the number of element(s) in set $A$ such that $^{7} C_{r} +^{7} C_{r - 1} =^{8} C_{s}$ is

A

8

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B

6

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C

4

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D

3

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Solution

The correct option is D $3$
Given : $^{7} C_{r} +^{7} C_{r - 1} =^{8} C_{s}$
$\Rightarrow^{8} C_{r} =^{8} C_{s}$
$(∵^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r})$

$\Rightarrow r + s = 8$ or $r = s$ but $r < s$
also $r - 1 \geq 0 \Rightarrow r \geq 1$
$∴$ Possible pairs of $(r, s) :$ $(1, 7), (2, 6), (3, 5)$
$∴$ Number of elements in set $A = 3$

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Standard XII Mathematics

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