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Permutations and Combinations

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Question : 19 of 56

Marks: +1, -0

If

{ }^{12} {C}_{5}+2{ }^{12} {C}_{4}+{ }^{12} {C}_{3}={ }^{14} {C}_{ {x}}

then the value of

{x}

5
9
5 or 9
None

Solution:

Concept:

Use Pascal's identity

{}^{n}C_{r-1}+{}^{n}C_{r}={}^{n+1}C_{r}

repeatedly to simplify the sum.

Explanation:

Group terms:

({}^{12}C_{5}+{}^{12}C_{4})+({}^{12}C_{4}+{}^{12}C_{3})

Apply identity:

{}^{12}C_{5}+{}^{12}C_{4}={}^{13}C_{5}

and

{}^{12}C_{4}+{}^{12}C_{3}={}^{13}C_{4}

So sum becomes

{}^{13}C_{5}+{}^{13}C_{4}

Apply identity again:

{}^{13}C_{5}+{}^{13}C_{4}={}^{14}C_{5}

Thus

{}^{14}C_{x}={}^{14}C_{5}

, so

x=5

or

x=14-5=9

(using

{}^{n}C_{r}={}^{n}C_{n-r}

).

Answer:

x = 5

or

9

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