Question

The value of \[^{{\mathbf{30}}}{{\mathbf{C}}_{{\mathbf{0}}}}^{{\mathbf{30}}}{{\mathbf{C}}_{{\mathbf{10}}}}{ - ^{{\mathbf{30}}}}{{\mathbf{C}}_{\mathbf{1}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{11}}}}{ + ^{{\mathbf{30}}}}{{\mathbf{C}}_{\mathbf{2}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{12}}}}.....{ + ^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{10}}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{20}}}}\;\] is:
A. $^{30}{C_{12}}$
B. $^{30}{C_{15}}$
C. $^{30}{C_{11}}$
D. $^{30}{C_{10}}$

Answer 1

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

Complete step-by-step answer:
We know that the binomial expansion of ${(1 + x)^n}$ , as
${(1 + x)^n}{ = ^n}{C_0}{ + ^n}{C_1}x{ + ^n}{C_2}{x^2} + .....{ + ^n}{C_n}{x^n}$ .
Substitute n = 30 in above expression, we get
${(1 + x)^{30}}{ = ^{30}}{C_0}{ + ^{30}}{C_1}x{ + ^{30}}{C_2}{x^2} + .....{ + ^{30}}{C_{30}}{x^{30}} …...….(1)$
Also, we know that the binomial expansion of ${(1 - x)^n}$ , as
${(1 - x)^n}{ = ^n}{C_0}{ - ^n}{C_1}x{ + ^n}{C_2}{x^2} - ..... + {( - 1)^r}{\;^n}{C_r}{x^n} + ....$ .
Substitute n = 30 in above expression, we get
${(1 - x)^{30}}{ = ^{30}}{C_0}{ - ^{30}}{C_1}x{ + ^{30}}{C_2}{x^2} - .....{ + ^{30}}{C_{30}}{x^{30}}…...….(2)$
Also, we know that
${(1 + x)^{30}}{(1 - x)^{30}} = {(1 - {x^2})^{30}}…...….(3)$
Thus the coefficient of ${x^{30}}$ in the equation on both sides will be equal.
Now, in binomial expansion of ${(1 - {x^2})^{30}}$i.e. RHS of equation (3) , coefficient ${x^{30}}$will be $^{30}{C_{10}}$ .
Also, in the multiplication of both expressions, means in LHS of equation (3), the coefficient of ${x^{30}}$ will be
\[^{{\mathbf{30}}}{{\mathbf{C}}_{{\mathbf{0}}}}^{{\mathbf{30}}}{{\mathbf{C}}_{{\mathbf{10}}}}{ - ^{{\mathbf{30}}}}{{\mathbf{C}}_{\mathbf{1}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{11}}}}{ + ^{{\mathbf{30}}}}{{\mathbf{C}}_{\mathbf{2}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{12}}}}.....{ + ^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{10}}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{20}}}}\;\],
Thus we get the comparison as:
\[^{{\mathbf{30}}}{{\mathbf{C}}_{{\mathbf{0}}}}^{{\mathbf{30}}}{{\mathbf{C}}_{{\mathbf{10}}}}{ - ^{{\mathbf{30}}}}{{\mathbf{C}}_{\mathbf{1}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{11}}}}{ + ^{{\mathbf{30}}}}{{\mathbf{C}}_{\mathbf{2}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{12}}}}.....{ + ^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{10}}}}{^{{\mathbf{30}}}}{{\mathbf{C}}_{{\mathbf{20}}}}\;\] = $^{30}{C_{10}}$
$\therefore $ Correct value will be $^{30}{C_{10}}$ .

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.