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Permutations and Combinations

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Question : 50 of 56

Marks: +1, -0

{n}+2 {C}_{n}=45

find the value of

{n}

[MTP-Oct-18]

7
8
9
6

Solution:

Concept:

Use the property

C(n+2, n) = C(n+2, 2)

and solve the quadratic.

Explanation:

We have:

{n+2}C_n = 45

.

Since

C(n+2, n) = C(n+2, 2)

, we get

C(n+2, 2) = 45

.

This gives

\frac{(n+2)(n+1)}{2} = 45

.

Multiply by 2:

(n+2)(n+1) = 90

.

Expand:

n^2 + 3n + 2 = 90 \Rightarrow n^2 + 3n - 88 = 0

.

Factor:

(n+11)(n-8)=0

, so

n = -11

or

n = 8

.

Since

n

is positive,

n = 8

.

Answer:

n = 8

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