(B) The given expression is $^{20}C_r \cdot ^{20}C_0 + ^{20}C_{r-1} \cdot ^{20}C_1 + \dots + ^{20}C_0 \cdot ^{20}C_r$.By Vandermonde's Identity, this

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(B) The given expression is $^{20}C_r \cdot ^{20}C_0 + ^{20}C_{r-1} \cdot ^{20}C_1 + \dots + ^{20}C_0 \cdot ^{20}C_r$.By Vandermonde's Identity, this sum is equivalent to the coefficient of $x^r$ in the expansion of $(1+x)^{20} \cdot (1+x)^{20} = (1+x)^{40}$.Thus, the sum is equal to $^{40}C_r$.The binomial coefficient $^{n}C_r$ is maximum when $r = n/2$.For $n = 40$, the maximum value occurs at $r = 40/2 = 20$.

Competitive Quantitative Aptitude Permutation and Combination Permutation and Combination

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The value of $r$ for which $^{20}C_r \cdot ^{20}C_0 + ^{20}C_{r-1} \cdot ^{20}C_1 + ^{20}C_{r-2} \cdot ^{20}C_2 + \dots + ^{20}C_0 \cdot ^{20}C_r$ is maximum is:

A
$15$
B
$20$
C
$11$
D
$10$

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There are $15$ buses running between Delhi and Mumbai. In how many ways can a man go to Mumbai and return by a different bus?

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Let $T_n$ denote the number of triangles which can be formed using the vertices of a regular polygon of $n$ sides. If $T_{n + 1} - T_n = 21$,then $n$ equals:

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How many words can be formed by using all the letters of the word '$TRIVANDRUM$' using each letter exactly once (in $!$)?

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$\sum\limits_{r = 0}^m {^{n + r}{C_n} = } $

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Six '$X$'s have to be placed in the squares of the figure such that each row contains at least one '$X$'. In how many different ways can this be done?

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The value of $r$ for which $^{20}C_r \cdot ^{20}C_0 + ^{20}C_{r-1} \cdot ^{20}C_1 + ^{20}C_{r-2} \cdot ^{20}C_2 + \dots + ^{20}C_0 \cdot ^{20}C_r$ is maximum is:

Similar Questions

There are $15$ buses running between Delhi and Mumbai. In how many ways can a man go to Mumbai and return by a different bus?

Let $T_n$ denote the number of triangles which can be formed using the vertices of a regular polygon of $n$ sides. If $T_{n + 1} - T_n = 21$,then $n$ equals:

How many words can be formed by using all the letters of the word '$TRIVANDRUM$' using each letter exactly once (in $!$)?

$\sum\limits_{r = 0}^m {^{n + r}{C_n} = } $

Six '$X$'s have to be placed in the squares of the figure such that each row contains at least one '$X$'. In how many different ways can this be done?

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