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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

Text Solution

Verified by Experts

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)

ML KHANNA-BINOMIAL THEOREM AND MATHEMATICAL INDUCTION -Problem Set (4) (MULTIPLE CHOICE QUESTIONS)
  1. If C(r ) stands for ""^(n)C(r ), then the sum of first (n+1) terms of...

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  2. The value of the sum of the series 3.""^(n)C(0)-8" "^(n)C(1)+13" "^(n...

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

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  4. If n is a positive integer and C(k)=""^(n)C(k), then the value of sum(...

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  5. The vaule of sum(r=0)^(n-1) (""^(C(r))/(""^(n)C(r) + ""^(n)C(r +1)) ...

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  6. sum(r=0)^(n) (-1)^(r )" "^(n)C(r ) (1+r x)/(1+n x) equals

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  7. If n gt 3, then abC(0)-(a-1) (b-1) C(1) + (a-2)(b-2) C(2)-(a-3) (b-3) ...

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  8. If C(r) be the coefficients of x^(r) in (1 + x)^(n) , then the value ...

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  9. If n is an odd natural number, then sum(r=0)^(n) ((-1)^(r ))/(""^(n)C(...

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  10. If a(n) = sum(r=0)^(n) (1)/(""^(n)C(r)) , find the value of sum(...

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  11. With usual notations C(0)C(1)+C(1)C(2)+…+C(n-1)C(n)=

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  12. With usual notations C(0)C(2)+C(1)C(3)+C(2)C(4)+…+C(n-2)C(n)=

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  13. With usual notations, C(0)C(r )+C(1)C(r+1) +C(2)C(r+2) +…+C(n-r)C(n)...

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  14. The coefficient of x^(n) y^(n) in the expansion of [(1 + x)(1+y) (x...

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  15. The coefficient of x^r[0lt=rlt=(n-1)] in lthe expansion of (x+3)^(n-1)...

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

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  17. The sum sum(i=0)^(m)""^(10)C(i)xx""^(20)C(m-i)("where " ""^(p)C(q)=0" ...

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  18. The value of of sum of the series ""^(14)C(0).""^(15)C(1)+""^(14)C(1)...

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  19. The value of (30 0)(30 10)-(30 1)(30 11)+(30 2)(30 12)++(30 20)(30 30)...

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  20. If C(r )=(""^(101)C(r )) then E=sum(r=0)^(100) (-1)^(r ) C(r ) C(r+1)

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