If one quarter of all the subsets containing three elements of the set A - a 1- a 2- a 3- - a n contains the element a 3- then the value of n isA- 10B- 12C- 14D- None of these

The correct option is B 12There are ^nC_3 subsets of set A, containing three elements. nbsp; If a_3 is always included, then there are ^n 1C_2 subsets of set ...

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Standard XII

Mathematics

Combination with Restrictions

If one quarte...

Question

If one quarter of all the subsets (containing three elements) of the set $A = {a_{1}, a_{2}, a_{3}, \dots, a_{n}}$ contains the element $a_{3},$ then the value of $n$ is

10

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12

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14

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None of these

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Solution

The correct option is B $12$
There are $^{n} C_{3}$ subsets of set $A,$ containing three elements.
If $a_{3}$ is always included, then there are $^{n - 1} C_{2}$ subsets of set $A .$
Given, $\frac{1}{4} \times^{n} C_{3} =^{n - 1} C_{2}$
$\Rightarrow \frac{n (n - 1) (n - 2)}{24} = \frac{(n - 1) (n - 2)}{2} \Rightarrow n = 12$

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Combination with Restrictions

Standard XII Mathematics

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If one quarter of all the subsets (containing three elements) of the set A={a1,a2,a3,…,an} contains the element a3, then the value of n is

The correct option is B 12There are nC3 subsets of set A, containing three elements. If a3 is always included, then there are n−1C2 subsets of set A. Given, 14×nC3=n−1C2 ⇒n(n−1)(n−2)24=(n−1)(n−2)2⇒n=12

If one quarter of all the subsets (containing three elements) of the set $A = {a_{1}, a_{2}, a_{3}, \dots, a_{n}}$ contains the element $a_{3},$ then the value of $n$ is