First slide

Binomial theorem for positive integral Index

Question

If $x + y = 1$ , then value of $\sum_{r = 0}^{n} (r) (^{n} C_{r}) x^{n - r} y^{r}$ is

Moderate

A

1
B

0
C

$n x$
D

$n y$

Solution

We have $(a + t)^{n} = \sum_{r = 0}^{n}^{n} C_{r} a^{n - r} t^{r}$
Differentiating both the sides with respect to t, we get
$n (a + t)^{n - 1} = \sum_{r = 0}^{n} r (^{n} C_{r}) a^{n - r} t^{r - 1}$
Multiplying both the sides by t and putting $a = x$ and $t = y$ , we get
$n (x + y)^{n - 1} y = \sum_{r = 0}^{n} r (^{n} C_{r}) x^{n - r} y^{r}$
$\Rightarrow \sum_{r = 0}^{n} r (^{n} C_{r}) x^{n - r} y^{r} = n y [∵ x + y = 1]$

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