For 1 - r - n - the value of nCr - n-1Cr- n-2Cr- rCr is -

The correct option is D n+1Cr+1 ^nC_r + ^n 1C_r + ^n 2C_r + .... + ^r+1C_r + ^rC_r = ^r+1C_r+1 + ^r+1C_r + ^r+2C_r + ^r+3C_r + ......+ ^n 1C_r + ^nC_r = ^r+2C ...

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Byju's Answer

Standard XI

Mathematics

Binomial Coefficients

For 1 ≤ r ≤ n...

Question

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

ⁿC_r+1

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ⁿ⁺¹C_r

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ⁿ⁺¹C_r+1

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None

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Solution

The correct option is C
ⁿ⁺¹C_r+1

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

= $^{r + 1} C_{r + 1} +^{r + 1} C_{r} +^{r + 2} C_{r} +^{r + 3} C_{r} + . . . . . . +^{n - 1} C_{r} +^{n} C_{r}$

= $^{r + 2} C_{r + 1} +^{r + 2} C_{r} +^{r + 3} C_{r} + . . . . . +^{n - 1} C_{r} +^{n} C_{r}$

= $^{r + 3} C_{r + 1} +^{r + 3} C_{r} + . . . . . +^{n - 1} C_{r} +^{n} C_{r}$

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

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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

Q. Prove that

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

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

Q. Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:
(a)

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

(b) n ·_{^{n −}}_¹C_r_{− 1} = (n − r + 1) ⁿC_r_{− 1}
(c)

\frac{{}^{n}C_{r}}{{}^{n - 1}C_{r - 1}} = \frac{n}{r}

(iv) ⁿC_r + 2 · ⁿC_r_{− 1} + ⁿC_r_{− 2} = ^{n +}²C_r.

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

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

The correct option is C n+1Cr+1 nCr+n−1Cr+n−2Cr+....+r+1Cr+rCr = r+1Cr+1+r+1Cr+r+2Cr+r+3Cr+......+n−1Cr+nCr = r+2Cr+1+r+2Cr+r+3Cr+.....+n−1Cr+nCr = r+3Cr+1+r+3Cr+.....+n−1Cr+nCr [nCr+nCr+1=n+1Cr+1]

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