The value of \(r\) for which \({ }^{20} \mathrm{C}_{\mathrm{r}}{ }^{20} \mathrm{C}_{0}+{ }^{20} \mathrm{C}_{\mathrm{r}-1}{ }^{20} \mathrm{C}_{1}+{ }^{20} \mathrm{C}_{\mathrm{r}-2}{ }^{20} \mathrm{C}_{2}+\ldots+{ }^{20} \mathrm{C}_{0}{ }^{20} \mathrm{C}_{\mathrm{r}}\) is maximum is:

JEE Main · 2019 · 11 Jan 2019 · Shift 1 · Binomial Theorem

The value of

r

for which

^{20} C_{r}^{20} C_{0} +^{20} C_{r - 1}^{20} C_{1} +^{20} C_{r - 2}^{20} C_{2} + \dots +^{20} C_{0}^{20} C_{r}

is maximum is:

Answer: (B)

The given sum is the Vandermonde convolution

\sum_{k = 0}^{r} (k 20) (r - k 20)

, which equals the coefficient of

x^{r}

(1 + x)^{20} (1 + x)^{20} = (1 + x)^{40}

, namely

(r 40)

. A binomial coefficient

(r 40)

is largest at the middle term,

r = 20

, so that is the value of

r

that maximises the sum.

JEE Main · 2019 · 11 Jan 2019 · Shift 1 · Single correct option