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Prove that "^nCr ^rC5= ^nC5 ^(n-5)C(r-5)...

Prove that `"^nC_r ^rC_5= ^nC_5 ^(n-5)C_(r-5)`.

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Prove that "^nP_r= ^nC_r ^rP_r .

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

Using Mathematical Induction, prove that : "^mC_0 ^nC_k+ ^mC_1 ^nC_(k-1)+..........+ ^mC_k ^nC_0= ^(m+n)C_k , where m, n, r are positive integers and "^pC_q = 0 for p < q.

Verify that ""^nC_r= frac (n)(r) "^(n-1)C_(r-1) and hence prove that "^nC_r= (n!)/(r!(n-r)!) .

Prove that sum_(r=0)^n 3^r "^nC_r= 4^n .

If m= "^nC_2 , prove that "^m C_2 =3xx ^(n+1)C_4 .

Prove that "^(n-1)C_3+ ^(n-1)C_4 > ^nC_3 if n >7.

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MODERN PUBLICATION-PERMUTATIONS AND COMBINATIONS-EXERCISE
  1. Prove that "^nCr+2 ^(n)C(r-1)+ ^(n)C(r-2) = ^(n+2)Cr.

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  2. Prove that "^nPr= ^nCr ^rPr.

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  3. Prove that "^nCr ^rC5= ^nC5 ^(n-5)C(r-5).

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  4. Prove that the product of any k consecutive integers is divisible by k...

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  5. Show that ((n+1)(n+2)(n+3)...... (n+r))/(r!) is a whole number.

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  6. Prove that "^(n-1)C3+ ^(n-1)C4 > ^nC3 if n >7.

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  7. Verify that ""^nCr= frac (n)(r) "^(n-1)C(r-1) and hence prove that "^n...

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  8. Find n if ""^nC4 , ""^nC5 and ""^nC6 are in A.P.

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  9. In how many ways can 5 sportsmen be selected from a group of 10 ?

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  10. How many selections of 4 books can be made from 8 different-books ?

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  11. A committee of 2 boys is to be selected from 4 boys. In how many ways ...

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  12. In how many ways can a committee be selected from 15 persons if the co...

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  13. In how many ways can a committee be selected from 15 persons if the co...

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  14. Sudha wants to choose any 9 stamps from a set of 11 different stamps. ...

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  15. How many lines can be drawn through 6 points on a circle ?

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  16. How many triangles can be drawn through n points on a circle ?

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  17. A polygon has 44 diagonals, find the number of its sides.

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  18. Find the number of diagonals of a : pentagon .

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  19. Find the number of diagonals of a : octagon .

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  20. If there are 12 persons in a party, and if each two of them shake hand...

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