If 1- x n - C 0- C 1 x - C 2 x 2- C n x n - n - N- then for 2 - m - n - C 0- C 1- C 2-1 m -1- C m -1 is equal toA- -1 m -1 n -1 C m -1B- n-1 Cm-1C- None of these D- -1 m n -1 C m -1

The correct option is A ( 1)^m 1 ^n 1C_m 1(1 x)^n(1 x)^ 1 ={C_0 C_1x+C_2x^2 ⋯+( 1)^m 1 C_m 1x^m 1 +⋯+( 1)^n C_nx^n}× {1+x+x^2+x^3+⋯+x^m 1+x^m+⋯} Equating ...

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Byju's Answer

Standard XII

Mathematics

Binomial Theorem for Any Index

If 1+ x n = C...

Question

If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{n} x^{n},$ $n \in N,$ then for $2 \leq m \leq n,$ $C_{0} - C_{1} + C_{2} - \dots + [(- 1)^{m - 1}] C_{m - 1}$ is equal to

(- 1)^{m - 1}^{n - 1} C_{m - 1}

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^{n - 1} C_{m - 1}

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(- 1)^{m}^{n - 1} C_{m - 1}

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None of these

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Solution

The correct option is A $(- 1)^{m - 1}^{n - 1} C_{m - 1}$
$(1 - x)^{n} (1 - x)^{- 1} = {C_{0} - C_{1} x + C_{2} x^{2} - \dots + (- 1)^{m - 1} C_{m - 1} x^{m - 1} + \dots + (- 1)^{n} C_{n} x^{n}} \times$
${1 + x + x^{2} + x^{3} + \dots + x^{m - 1} + x^{m} + \dots}$

Equating coefficients of $x^{m - 1}$ on both sides, we get
$(- 1)^{m - 1}^{n - 1} C_{m - 1} = C_{0} - C_{1} + C_{2} - \dots + [(- 1)^{m - 1}] C_{m - 1}$

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If (1+x)n=C0+C1x+C2x2+⋯+Cnxn, n∈N, then for 2≤m≤n, C0−C1+C2−⋯+[(−1)m−1]Cm−1 is equal to

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If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{n} x^{n},$ $n \in N,$ then for $2 \leq m \leq n,$ $C_{0} - C_{1} + C_{2} - \dots + [(- 1)^{m - 1}] C_{m - 1}$ is equal to