NCERT solution Class 11 binomial theorem

NCERT Solutions

Class 11 Maths

Binomial Theorem

NCERT Questions

Ex.8.1 Q.1
Ex.8.1 Q.2
Ex.8.1 Q.3
Ex.8.1 Q.4
Ex.8.1 Q.5
Ex.8.1 Q.6
Ex.8.1 Q.7
Ex.8.1 Q.8
Ex.8.1 Q.9
Ex.8.1 Q.10
Ex.8.1 Q.11
Ex.8.1 Q.12
Ex.8.1 Q.13
Ex.8.1 Q.14
Ex.8.2 Q.1
Ex.8.2 Q.2
Ex.8.2 Q.3
Ex.8.2 Q.4
Ex.8.2 Q.5
Ex.8.2 Q.6
Ex.8.2 Q.7
Ex.8.2 Q.8
Ex.8.2 Q.9
Ex.8.2 Q.10
Ex.8.2 Q.11
Ex.8.2 Q.12
Ex.Misc Q.1
Ex.Misc.Q.2
Ex.Misc.Q.3
Ex.Misc.Q.4
Ex.Misc.Q.5
Ex.Misc.Q.6
Ex.Misc.Q.7
Ex.Misc.Q.8
Ex.Misc.Q.9
Ex.Misc.Q.10

Ex.8.2 Q.9

In the expansion of (1 + a)^{m + n}, prove that coefficients of a^m and aⁿ are equal.

It is known that (r + 1)^th term, (T_r+1), in the binomial expansion of (a + b)ⁿ is given by

T_r+1 = ⁿC_r a^n-r b^r

Assuming that a^m occurs in the (r + 1)^th term of the expansion (1 + a)^{m + n}, we obtain

T_r+1 = ^m+nC_r (1)^m+n-r a^r

Comparing the indices of a in a^m and in T_{r + 1}, we obtain

r = m

Therefore, the coefficient of a^m is

^m+nC_m = (m + n)! ÷ {m! × (m +n - m)!} = (m + n)! ÷ (m! × n!) …………… (1)

Assuming that aⁿ occurs in the (k + 1)^th term of the expansion (1 + a)^m+n, we obtain

T_k+1 = ^m+nC_k (1)^m+n-k a^k= ^m+nC_k a^k

Comparing the indices of a in aⁿ and in T_{k + 1}, we obtain

k = n

Therefore, the coefficient of aⁿ is

^m+nC_n = (m + n)! ÷ {n! × (m +n - n)!} = (m + n)! ÷ (n! × m!) …………… (2)

Thus, from equation 1 and 2, it can be observed that the coefficients of a^m and aⁿ

in the expansion of (1 + a)^{m + n} are equal.

Video solution:

Binomial Theorem Class 11 Maths NCERT Chapter 7 NCERT Solutions

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NCERT solutions of Binomial Theorem Class 11 Maths - all questions of all exercises are solved. Q. No. 1, 2, 3, and so on. Complete explanation and detailed solution makes them the best videos for Class 11 Maths Binomial Theorem NCERT solutions.