The value of $^{n} C_{1} +^{n} C_{2} +^{n} C_{3} + . . . . . +^{n} C_{n}$ is $2^{n}$

True

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False

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Solution

The correct option is B False
We know that

$(1 + x)^{n} =^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} + . . . . . +^{n} C_{n} x^{n}$

Put $x = 1$

Then,
$(1 + 1)^{n} =^{n} C_{0} +^{n} C_{1} \times 1 +^{n} C_{2} \times 1^{2} + . . . . . +^{n} C_{n} \times 1^{n}$

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

$^{n} C_{1} +^{n} C_{2} + . . . . . +^{n} C_{n} = 2^{n} - 1$

Hence, this is the answer.

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The value of nC1 + nC2 + nC3 +.....+ nCn is 2n

The correct option is B FalseWe know that(1+x)n=nC0+nC1x+nC2x2+.....+nCnxnPut x=1Then,(1+1)n=nC0+nC1×1+nC2×12+.....+nCn×1n2n=1+nC1+nC2+.....+nCnnC1+nC2+.....+nCn=2n−1Hence, this is the answer.

The value of $^{n} C_{1} +^{n} C_{2} +^{n} C_{3} + . . . . . +^{n} C_{n}$ is $2^{n}$