Let r and n be positive integers such that 1 - r - n- Then prove the following -i n C T - n C r -1- n - r -1- rii nn-1 Cr-1-n-r-1n Cr-1iii n C r - n -1 C r -1- n - riv n C r -2 n C r -1- n C r -2- n -2 C r

(i) ^nC_r/^nC_r 1 = n r+1/r As we knows ^nC_r = n!/r!(n 2)! ∴^nC_r 1 = n!/(r 1)!(n r + 1)! ^nC_r/^nC_r 1 = n!(r 1)!(n r+1)!/r!(n r)! n! = (r 1)!(n r+1) × (n r) ...

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Byju's Answer

Standard XII

Mathematics

Sum of Coefficients of All Terms

Let r and n b...

Question

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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Solution

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

As we knows $^{n} C_{r} = \frac{n!}{r! (n - 2)!}$

$∴^{n} C_{r - 1} = \frac{n!}{(r - 1)! (n - r + 1)!} \frac{^{n} C_{r}}{^{n} C_{r - 1}} = \frac{n! (r - 1)! (n - r + 1)!}{r! (n - r)! n!} = \frac{(r - 1)! (n - r + 1) \times (n - r)!}{r_{2} \times (r - 1)! (n - r)!} = \frac{n - r + 1}{r}$
Hence proved.
(ii) $n^{n - 1} C_{r - 1} = (n - r + 1)^{n} C_{r - 1}$
$n \times^{n - 1} C_{r - 1} = n \times \frac{(n - 1)!}{(r - 1)! [(n - 1) - (r - 1)]!} = \frac{n! \times (n - r + 1)}{(r - 1)! (n - r)! (n - r + 1)}$
multiplying numerator and denominator by (n-r+1)]
$= \frac{(n - r + 1) \times n!}{(r - 1)! (n - r + 1)!} = (n - r + 1)^{n} C_{r - 1}$
Hence proved.

(iii) $\frac{^{n} C_{r}}{^{n - 1} C_{r - 1}} = \frac{n}{r}$
$^{n} C_{r} = \frac{n!}{r! (n - r)!}^{n - 1} C_{r - 1} = \frac{(n - 1)!}{(r - 1)! (n - 1) - (r - 1)!} O r \frac{^{n} C_{r}}{^{n - 1} C_{r - 1}} = \frac{n! (r - 1)! (n - r)!}{r! (n - r! (n - 1)!} = \frac{n \times (n - 1)! (r - 1)! \times (n - r)!}{r \times (n - 1)! (r - 1)! \times (n - r)!} = \frac{n}{r}$

Hence proved.

(iv) $^{n} C_{r} +^{2 n} C_{r - 1} +^{n} C_{r - 2} =^{n + 2} C_{r} .$
$L . H . S . \Rightarrow^{n} C_{r} +^{2 n} C_{r - 1} +^{n} C_{r - 2} = (^{n} C_{r} +^{n} C_{r - 1}) + (^{n} C_{r - 2} +^{n} C_{r - 1}) =^{n + 1} C_{r} +^{n + 1} C_{r - 1} [∵^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}] = (n + 1) +^{1} C_{r} =^{n + 2} C_{r}$

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Similar questions

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.

Q. If

(2 \leq r \leq n),

then

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

is equal to

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

Q. Prove that

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

when

1 \leq r \leq n

Q. Prove that

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

where

1 \leq r \leq n

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Let r and n be positive integers such that 1≤r≤n. Then prove the following : (i) nCrnCr−1=n−r+1r (ii) nn−1Cr−1=(n−r+1)nCr−1 (iii) nCrn−1Cr−1=nr (iv) nCr+2nCr−1+nCr−2=n+2Cr