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

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Solution

L.H. S. = $^{n} C_{r} +^{n} C_{r - 1} +^{n} C_{r - 1} +^{n} C_{r - 2}$

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

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Similar questions

Q. Prove that

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

Q. If

(2 \leq r \leq n),

then

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

is equal to

Q. Prove that

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

Q. Prove that

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

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

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Prove that nCr+2.nCr−1+nCr−2=n+2Cr.

L.H. S. = nCr+nCr−1+nCr−1+nCr−2 = n+1Cr+n+1Cr−1=n+2Cr.

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

L.H. S. = $^{n} C_{r} +^{n} C_{r - 1} +^{n} C_{r - 1} +^{n} C_{r - 2}$

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