The evaluated value of -i-0n-j-1n nCj jCi- i- j

The correct option is A 3^n 1 Expanding, we get nbsp; ^nC_1( ^1C_0+ ^1C_1)+ ^nC_2( ^2C_0+ ^2C_1+ ^2C_2)+.... ^nC_n( ^nC_1+ ^nC_2+... ^nC_n) = ^ ...

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Standard XII

Mathematics

Binomial Coefficients

The evaluated...

Question

The evaluated value of $n \sum i = 0 n \sum j = 1$ $^{n} C_{j}$ $^{j} C_{i},$ $i \leq j$

3^{n} + 1

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3^{n} - 1

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3^{n + 1} + 1

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None of these

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Solution

The correct option is A $3^{n} - 1$
Expanding, we get
$^{n} C_{1} (^{1} C_{0} +^{1} C_{1}) +^{n} C_{2} (^{2} C_{0} +^{2} C_{1} +^{2} C_{2}) + . . . .^{n} C_{n} (^{n} C_{1} +^{n} C_{2} + . . .^{n} C_{n})$
$=^{n} C_{1} 2^{1} +^{n} C_{2} 2^{2} + . . .^{n} C_{n} 2^{n}$
$=^{n} C_{0} 2^{0} +^{n} C_{1} 2^{1} +^{n} C_{2} 2^{2} + . . .^{n} C_{n} 2^{n} - [^{n} C_{0} 2^{0}]$
$= (1 + 2)^{n} - 1$
$= 3^{n} - 1$

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The evaluated value of n∑i=0n∑j=1 nCj jCi, i≤j

The correct option is A 3n−1Expanding, we get nC1(1C0+1C1)+nC2(2C0+2C1+2C2)+....nCn(nC1+nC2+...nCn)=nC121+nC222+...nCn2n=nC020+nC121+nC222+...nCn2n−[nC020]=(1+2)n−1=3n−1

The evaluated value of $n \sum i = 0 n \sum j = 1$ $^{n} C_{j}$ $^{j} C_{i},$ $i \leq j$