A set contains elements. If the number of subsets which contain at most elements is 4096, then the value of is:

6 Explanation: 1. Understand the Problem Statement The total number of elements in the set is . We are given that the number of subsets containing at most elements is 4096. This can be expressed as the sum of binomial coefficients: 2. Use Binomial Symmetry Property We know the property of binomial coefficients: . Therefore, in a set with elements: ... 3. Relate to the Total Number of Subsets The total number of subsets for a set with elements is . The sum of all binomial coefficients is: Because of symmetry, the two groups in the parentheses are equal. Let the sum be : 4. Solve for We are given : We know that (since and ): Comparing the powers: Correct Option: (d) 6

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NIMCET 2009 — Mathematics PYQ

NIMCET | Mathematics | 2009

A set contains $(2n + 1)$ elements. If the number of subsets which contain at most $n$ elements is 4096, then the value of $n$ is:

Choose the correct answer:

Explanation

1. Understand the Problem Statement

The total number of elements in the set is $N = 2n + 1$ .

We are given that the number of subsets containing at most $n$ elements is 4096.

This can be expressed as the sum of binomial coefficients:

$^{2n+1}C_0 + ^{2n+1}C_1 + ^{2n+1}C_2 + \dots + ^{2n+1}C_n = 4096$

2. Use Binomial Symmetry Property

We know the property of binomial coefficients: $^{n}C_r = ^{n}C_{n-r}$ .

Therefore, in a set with $2n+1$ elements:

$^{2n+1}C_0 = ^{2n+1}C_{2n+1}$

$^{2n+1}C_1 = ^{2n+1}C_{2n}$

...

$^{2n+1}C_n = ^{2n+1}C_{n+1}$

3. Relate to the Total Number of Subsets

The total number of subsets for a set with $2n+1$ elements is $2^{2n+1}$ .

The sum of all binomial coefficients is:

$(^{2n+1}C_0 + \dots + ^{2n+1}C_n) + (^{2n+1}C_{n+1} + \dots + ^{2n+1}C_{2n+1}) = 2^{2n+1}$

Because of symmetry, the two groups in the parentheses are equal. Let the sum be $S$ :

$S + S = 2^{2n+1}$

$2S = 2^{2n+1}$

$S = 2^{2n}$

4. Solve for $n$

We are given $S = 4096$ :

$2^{2n} = 4096$

We know that $4096 = 2^{12}$ (since $2^{10} = 1024$ and $1024 \times 4 = 4096$ ):

$2^{2n} = 2^{12}$

Comparing the powers:

$2n = 12$

$n = 6$

Correct Option: (d) 6