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If m,n,r are positive integers such that...

If m,n,r are positive integers such that r `lt ` m,n, then
`""^(m)C_(r)+""^(m)C_(r-1)""^(n)C_(1)+""^(m)C_(r-2)""^(n)C_(2)+...+ ""^(m)C_(1)""^(n)C_(r-1)+""^(n)C_(r)` equals

A

`(""^(n)C_(r))^(2)`

B

`""^(m+n)C_(r)`

C

`""^(m+n)C_(r) + ""^(m)C_(r) + ""^(n)C_(r)`

D

none of these

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The correct Answer is:
B
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""^(n-2)C_(r)+2""^(n-2)C_(r-1)+""^(n-2)C_(r-2) equals :

""^(n)C_(r+1)+^(n)C_(r-1)+2.""^(n)C_(r)=

""^(n)C_(r)+2""^(n)C_(r-1)+C_(r-2) is equal to

Prove that ""^(n)C_(r )+2""^(n)C_(r-1)+ ""^(n)C_(r-2)= ""^(n+2)C_(r ) .

Let m, in N and C_(r) = ""^(n)C_(r) , for 0 le r len Statement-1: (1)/(m!)C_(0) + (n)/((m +1)!) C_(1) + (n(n-1))/((m +2)!) C_(2) +… + (n(n-1)(n-2)….2.1)/((m+n)!) C_(n) = ((m + n + 1 )(m+n +2)…(m +2n))/((m +n)!) Statement-2: For r le 0 ""^(m)C_(r)""^(n)C_(0)+""^(m)C_(r-1)""^(n)C_(1) + ""^(m)C_(r-2) ""^(n)C_(2) +...+ ""^(m)C_(0)""^(n)C_(r) = ""^(m+n)C_(r) .

Let n and r be no negative integers suych that r<=n. Then,^(n)C_(r)+^(n)C_(r-1)=^(n+1)C_(r)

OBJECTIVE RD SHARMA-BINOMIAL THEOREM AND ITS APPLCIATIONS -Exercise
  1. " if" C(0)C(1)C(2),……C(n) are the binomial coefficients in the exp...

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  2. Find the sum 2C0(23)/2C1+(2^3)/3C2+(2^4)/4C3++(2^(11))/(11)C(10)dot

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  3. If m,n,r are positive integers such that r lt m,n, then ""^(m)C(r)...

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  4. Find the value of 1/(81^n)-(10)/(81^n)^(2n)C1+(10^2)/(81^n)^(2n)C2-(10...

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  5. The term independent of x in the expansion of (x-1/x)^4(x+1/x)^3 is

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  6. if the coefficients of x^(5)" and "x^(15) in the expansion of (x^(2)+(...

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  7. If n is a positive integer and Ck = .^nCk then find the value of sum(k...

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  8. Find the coefficients of x^(50) in the expression (1+x)^(1000)+2x(1+x)...

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  9. If Cr stands for .^nCr, then the sum of the series (2(n/2)!(n/2)!)/...

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  10. The coefficient of x^(n) in the expansion of (1)/((1-x)(3 -x)), is

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  11. The coefficient of x^(n) in the expansion of ((1+x)^(2))/((1 - x)^(3))...

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  12. Coefficient of x^n in the expansion of (1-2x+3x^2-4x^3+....oo)^2 is

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  13. If (r+1)t h term is the first negative term in the expansion of (1+x)^...

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  14. The coefficient of x^6 in the expansion of (1+x+x^2)^(-3), is

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  15. The coefficient of x^n in the expansion of (1 + x+x^2+...........)^-...

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  16. If the binomial expansion of (a +b x)^-2 is 1/4-3x+......., then (a, ...

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  17. If C(r) = ""^(n)C(r) and (C(0) + C(1)) (C(1) + C(2)) … (C(n-1) + C(n))...

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  18. If the third term in the expansion of (1+x)^mi s-1/8x^2, then find the...

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  19. If p is nearly equal to q and n gt 1 , such that ((n+1) p+(n-1)q)/(...

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  20. If y=3 x + 6 x^(2) + 10 x^(3) +… then x =

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