If $^{n} C_{r}$ denotes the number of combinations of $n$ things taken $r$ at a time, then the expression $^{n} C_{r + 1} +^{n} C_{r - 1} + 2 \times^{n} C_{r}$ equals

^{n + 2} C_{r}

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^{n + 2} C_{r + 1}

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^{n + 1} C_{r}

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^{n + 1} C_{r + 1}

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Solution

The correct option is B $^{n + 2} C_{r + 1}$
$^{n} C_{r + 1} +^{n} C_{r - 1} + 2 \times^{n} C_{r}$

= $^{n} C_{r + 1} +^{n} C_{r} +^{n} C_{r - 1} +^{n} C_{r}$ $∵$ $^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$

= $^{n + 1} C_{r + 1} +^{n + 1} C_{r}$

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

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Similar questions

Q. If

(2 \leq r \leq n),

then

^{n} C_{r} + 2 \cdot^{n} C_{r + 1} +^{n} C_{r + 2}

is equal to

Let r and n be positive integers such that $1 \leq r \leq n .$ Then prove the following :

(i) $\frac{^{n} C_{r}}{^{n} C_{r - 1}} = \frac{n - r + 1}{r}$
(ii) $n^{n - 1} C_{r - 1} = (n - r + 1)^{n} C_{r - 1}$
(iii) $\frac{^{n} C_{r}}{^{n - 1} C_{r - 1}} = \frac{n}{r}$
(iv) $^{n} C_{r} + 2^{n} C_{r - 1} +^{n} C_{r - 2} =^{n + 2} C_{r}$

For $1 \leq r \leq n$ , the value of $ {}^nC_r +^{{}^{n-1}C_r}+^{{}^{n-2}C_r}.....+...+....+^{{}^rC_r} \space is : $

Q. If

^{n} C_{r}

denots the number of combinations of

n

things taken

r

at a time, then the expression

^{n} C_{r + 1} +^{n} C_{r - 1} + 2 \times^{n} C_{r}

equals

Q. The expression

^{n} C_{r} + 2^{n} C_{r - 1} +^{n} C_{r - 2}

is equal to

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If nCr denotes the number of combinations of n things taken r at a time, then the expression nCr+1+nCr−1+2×nCr equals

The correct option is B n+2Cr+1nCr+1+nCr−1+2×nCr= nCr+1+nCr+nCr−1+nCr ∵ nCr+nCr−1=n+1Cr= n+1Cr+1+n+1Cr= n+2Cr+1

If $^{n} C_{r}$ denotes the number of combinations of $n$ things taken $r$ at a time, then the expression $^{n} C_{r + 1} +^{n} C_{r - 1} + 2 \times^{n} C_{r}$ equals

The correct option is B $^{n + 2} C_{r + 1}$
$^{n} C_{r + 1} +^{n} C_{r - 1} + 2 \times^{n} C_{r}$

= $^{n} C_{r + 1} +^{n} C_{r} +^{n} C_{r - 1} +^{n} C_{r}$ $∵$ $^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$

= $^{n + 1} C_{r + 1} +^{n + 1} C_{r}$

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