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Question: The value of \[^{{\mathbf{30}}}{{\mathbf{C}}_{{\mathbf{0}}}}^{{\mathbf{30}}}{{\mathbf{C}}_{{\mathbf{...

The value of 30C030C10−30C130C11+30C230C12.....+30C1030C20  ^{{\mathbf{30}}}{{\mathbf{C}}_{{\mathbf{0}}}}^{{\mathbf{30}}}{{\mathbf{C}}_{{\mathbf{10}}}}{ - ^{{\mathbf{30}}}}{{\mathbf{C}}_{\mathbf{1}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{11}}}}{ + ^{{\mathbf{30}}}}{{\mathbf{C}}_{\mathbf{2}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{12}}}}.....{ + ^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{10}}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{20}}}}\; is:
A. 30C12^{30}{C_{12}}
B. 30C15^{30}{C_{15}}
C. 30C11^{30}{C_{11}}
D. 30C10^{30}{C_{10}}

Explanation

Solution

This is based on the binomial expansion of (1+x)n{(1 + x)^n} and (1−x)n{(1 - x)^n} . Expand it in terms of combinatorial values. Then have the binomial expansion of (1−x2)30{(1 - {x^2})^{30}}. Compare the coefficients of x30{x^{30}} in the product of (1+x)n{(1 + x)^n} and (1−x)n{(1 - x)^n}with (1−x2)30{(1 - {x^2})^{30}}. This comparison will give the required result.

Complete step-by-step answer:
We know that the binomial expansion of (1+x)n{(1 + x)^n} , as
(1+x)n=nC0+nC1x+nC2x2+.....+nCnxn{(1 + x)^n}{ = ^n}{C_0}{ + ^n}{C_1}x{ + ^n}{C_2}{x^2} + .....{ + ^n}{C_n}{x^n} .
Substitute n = 30 in above expression, we get
(1+x)30=30C0+30C1x+30C2x2+.....+30C30x30…...….(1){(1 + x)^{30}}{ = ^{30}}{C_0}{ + ^{30}}{C_1}x{ + ^{30}}{C_2}{x^2} + .....{ + ^{30}}{C_{30}}{x^{30}} …...….(1)
Also, we know that the binomial expansion of (1−x)n{(1 - x)^n} , as
(1−x)n=nC0−nC1x+nC2x2−.....+(−1)r  nCrxn+....{(1 - x)^n}{ = ^n}{C_0}{ - ^n}{C_1}x{ + ^n}{C_2}{x^2} - ..... + {( - 1)^r}{\;^n}{C_r}{x^n} + .... .
Substitute n = 30 in above expression, we get
(1−x)30=30C0−30C1x+30C2x2−.....+30C30x30…...….(2){(1 - x)^{30}}{ = ^{30}}{C_0}{ - ^{30}}{C_1}x{ + ^{30}}{C_2}{x^2} - .....{ + ^{30}}{C_{30}}{x^{30}}…...….(2)
Also, we know that
(1+x)30(1−x)30=(1−x2)30…...….(3){(1 + x)^{30}}{(1 - x)^{30}} = {(1 - {x^2})^{30}}…...….(3)
Thus the coefficient of x30{x^{30}} in the equation on both sides will be equal.
Now, in binomial expansion of (1−x2)30{(1 - {x^2})^{30}}i.e. RHS of equation (3) , coefficient x30{x^{30}}will be 30C10^{30}{C_{10}} .
Also, in the multiplication of both expressions, means in LHS of equation (3), the coefficient of x30{x^{30}} will be
30C030C10−30C130C11+30C230C12.....+30C1030C20  ^{{\mathbf{30}}}{{\mathbf{C}}_{{\mathbf{0}}}}^{{\mathbf{30}}}{{\mathbf{C}}_{{\mathbf{10}}}}{ - ^{{\mathbf{30}}}}{{\mathbf{C}}_{\mathbf{1}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{11}}}}{ + ^{{\mathbf{30}}}}{{\mathbf{C}}_{\mathbf{2}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{12}}}}.....{ + ^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{10}}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{20}}}}\;,
Thus we get the comparison as:
30C030C10−30C130C11+30C230C12.....+30C1030C20  ^{{\mathbf{30}}}{{\mathbf{C}}_{{\mathbf{0}}}}^{{\mathbf{30}}}{{\mathbf{C}}_{{\mathbf{10}}}}{ - ^{{\mathbf{30}}}}{{\mathbf{C}}_{\mathbf{1}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{11}}}}{ + ^{{\mathbf{30}}}}{{\mathbf{C}}_{\mathbf{2}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{12}}}}.....{ + ^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{10}}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{20}}}}\; = 30C10^{30}{C_{10}}
∴\therefore Correct value will be 30C10^{30}{C_{10}} .

So, the correct answer is “Option D”.

Note: The Binomial Theorem is a faster method for expanding (or multiplying out) a binomial expression with some exponent value. Any coefficient of the terms in expansion are represented by combinatorial terms. Such coefficients are known as binomial coefficient. Obviously such binomial coefficients are useful in combinatorics problems. It gives the number of different combinations of b elements that can be chosen from a set of n elements.This question is based on the concept that coefficients of the terms with same exponents of variables in expansion must be equal in any equation given.