For n-3-1-2 n C r -2-3 n C r -1- - -1 r r -1 r -2 isA- n-3 CrB- n-3 CrC- n-3 Cr-1D- n-2 Cr

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Sum of Product of Binomial Coefficients

For n>3,1.2 n...

Question

For n >3 , ${1.2}^{n} C_{r} - {2.3}^{n} C_{r - 1} + . . . . . + (- 1)^{r} (r + 1) (r + 2)$ is

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B
we have $(1 + x)^{n} =^{n} C_{0} +^{n} C_{1} +^{n} C_{2} x^{2} + . . . . . .$
$+^{n} C_{r - 1} x^{r - 1} +^{n} C_{r} x^{r} + . . . . +^{n} C_{n} x^{n} . . . . (1)$
and $(1 + x)^{- 3} = 1 -^{3} C_{1} x +^{4} C_{2} x^{2} . . . . . +$ $(- 1)^{r - 1}^{r + 1} C_{r - 1} x^{r - 1} + (- 1)^{r}$
$^{r + 2} C_{r} x^{r} + . . . . . . t o \infty . . . . . (2)$
Multiply (1) and (2), we get
$(1 + x)^{n - 3} = (^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} + . . . . +^{n} C_{n} x^{n}) (1 -^{3} C_{1} x +^{4} C_{2} x^{2} - . . . \infty)$
Clearly , the coefficient of $x^{r}$ from the product in R.H.S is
1. $^{n} C_{r} -^{3} C_{1} .^{n} C_{r - 1} +^{4} C_{2} .^{n} C_{r - 2} - . . . . + (- 1)^{r}$ $^{r + 2} C_{r} .^{n} C_{0}$
$=^{n} C_{r} - 3^{n} C_{1} .^{n} C_{r - 1} + {\frac{4.3}{2.1}}^{n} C_{r - 2} - {\frac{5.4}{2.1}}^{n} C_{r - 3}$
$+ . . . + (1)^{r} \frac{(r + 2) (r + 1)}{2.1}$
$= \frac{1}{2} [{1.2}^{n} C_{r} - {2.3}^{n} C_{r - 1} + {4.3}^{n} C_{r - 2} - . . . . + (- 1) r (r + 2) (r + 1)]$
$∴$ Required series $= 2 \times$ coefficient of $x^{r}$ in $(1 + x)^{n - 3} = {2.}^{n - 3} C_{r}$

Suggest Corrections

Similar questions

Q. Sum of series

\sum_{r = 1}^{n} (r^{2} + 1) r!

Which of the following best describes the terms of the series? $\frac{1}{2}$ $^{n} C_{1}$ - $\frac{2}{3}$ $^{n} C_{2}$ + $\frac{3}{4}$ $^{n} C_{3}$ ................... $(- 1)^{n + 1}$ $\frac{1}{n + 1}$ $^{n} C_{n}$

Q. The coefficient of

x^{n}

in the expansion of

(1 - 3 x + 6 x^{2} - 10 x^{3} + . . . + (- 1)^{r} \frac{(r + 1) (r + 2) x^{r}}{2} + . . . \infty)^{- n}

Q. The value of

lim x \to \infty \frac{1 \cdot n \sum r = 1 r + 2 \cdot n - 1 \sum r = 1 r + 3 \cdot n - 2 \sum r = 1 r + . . . . + n \cdot 1}{n^{4}}

Q. For

n \in N

. let

S (n) = \sum_{r = 0}^{n} {(- 1)}^{r} \frac{1}{({^{n} C}_{r})}

Value of

S = \sum_{r = 0}^{n} {(- 1)}^{r} \frac{({^{r + 2} C}_{r})}{({^{n} C}_{r})}

Join BYJU'S Learning Program

For n >3 ,1.2nCr−2.3nCr−1+.....+(−1)r(r+1)(r+2) is

For n >3 , ${1.2}^{n} C_{r} - {2.3}^{n} C_{r - 1} + . . . . . + (- 1)^{r} (r + 1) (r + 2)$ is