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Show that the coefficient of a^m and a^n...

Show that the coefficient of `a^m` and `a^n` in expansion of `(1+a)^(m+n)` are equal.

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(m+1)th and (n+1)th term in the expansion of
`(1+a)^(m+n)` are `"^(m+n)C_m a^m` and `^(m+n)C_n a^n`
`therefore` The coefficient of `a^m` and `a^n` are `"^(m+n)C_m` and `^(m+n)C_n` which are equal.
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MBD PUBLICATION-Elements of Mathematics-QUESTION BANK
  1. Find the coefficient of x^4 in the expansion of (1+3x+10x^2)(x+1/x)^10

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  2. Find the term independent of x in the above expansion. (1+3x+10x^2)(x+...

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  3. Show that the coefficient of a^m and a^n in expansion of (1+a)^(m+n) a...

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  4. An expression of the form (a+b+c+d+ .... ) consisting of sum of many d...

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  5. State and prove a multinomial Theorem.

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  6. Prove that "^(2n)C0 + ^(2n)C2 + .... + ^(2n)C(2n) = 2^(2n-1)

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  7. Prove that "^(2n)C1 + ^(2n)C3 + .... + ^(2n)C(2n-1) = 2^(2n-1)

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  8. Find the sum of C1 + 2C2 + 3C3 + .... + nCn

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  9. Find the sum of C0 + 2C1 + 3C2 + .... + (n+1)Cn

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  10. Compute ((1+k)(1+k/2) ..... (1+k/n))/((1+n)(1+n/2) ..... (1+n/k))

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  11. Show that C0 C1 + C1 C2 + C2 C3 + .... + C(n-1) Cn = (2n!)/((n-1)!(n...

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  12. C0 C1 + C1 C2 + .... + C(n-1) Cn = (2^n.n.1.3.5... (2n-1))/(n+1)

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  13. Show that 3C0-8C1 + 13C2 - 18C3 + ..... + (n+1)^(th) term = 0

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  14. Show that C0 n^2 + C1 (2-n)^2 + C2 (4-n)^2 + .... + Cn (2n-n)^2 = n.2^...

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  15. Show that C0 + 3C1 + 5C2 + .... +(2n+1) Cn = (n+1)(2^n)

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  16. Find the sum of the following C1 - 2C2 + 3C3 - ..... + n(-1)^(n-1) Cn

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  17. Find the sum of the following 1.2 C(2) + 2.3 C(3) + ... + (n-1)nCn

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  18. Find the sum of the following C1 + 2^2 C2 + 3^2 C3 + ... + n^2Cn

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  19. Find the sum of the following C1 - 2C2 + 3C3 - ..... + n(-1)^(n-1) Cn

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  20. Show that C1^2 + 2C2^2 + 3C3^2 + ... + "^nCn^2 = ((2n-1!))/{(n-1)!}^2

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