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If (1 + x)^(n) = C(0) + C(1) x + C(2) ...

If ` (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n)` , prove that
` C_(0) *""^(2n)C_(n) - C_(1) *""^(2n-2)C_(n) + C_(2) *""^(2n-4) C_(n) -…= 2^(n)`

Text Solution

Verified by Experts

The correct Answer is:
A, B
`(a,b)` The given series can be written as
`S=sum_(r=0)^(n)'^(n)C_(r )^(2n-2)C_(m)(-1)^(r )`
`=sum_(r=0)^(n)'^(n)C_(r )(-1)^(r )xx"coefficient of" x^(m) "in" (1+x)^(2n-2r)`
`="coefficient of" x^(m) "in" sum_(r=0^(n)'^(n)C_(r )(-1)^(r )[(1+x)^(2)]^(n-r)`
`="coefficient of" x^(m) "in" (x^(2)+2x)^(n)`
`="coefficient of" x^(m) "in" x^(n) (x+2)^(n)`
`="coefficient of" x^(m-n) "in" (x+2)^(n)`
`=^(n)C_(m-n)2^(n-(m-n))=((n),(m-n))2^(2n-m)` if `m ge n` and `0` if `m lt n`.
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Prove that ^nC_0 "^(2n)C_n-^nC_1 ^(2n-2)C_n +^nC_2 ^(2n-4)C_n =2^n

If (1+x)^(n)=^(n)C_(0)+^(n)C_(1)x+^(n)C_(2)x^(2)+…+^(n)C_(n)x^(n) , prove that, nC_(1)-2^(n)C_(2)+3^(n)C_(3)-…+(-1)^(n-1).n^(n)C_(n)=0 .

Knowledge Check

  • The value of (""^(n) C_(1)+2. ""^(n) C_(2)+3. ""^(n)C_(3)+…+ n""^(n)C_(n)) is-

    A
    `2^(n)`
    B
    `n.2 ^(n)`
    C
    `n. 2^(n-1)`
    D
    `n. 2^(n+1)`

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