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If m= "^nC2, prove that "^m C2 =3xx ^(n...

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

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

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

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

MODERN PUBLICATION-PERMUTATIONS AND COMBINATIONS-EXERCISE
  1. If .^(n)C(r )r: ^(n)C(r+1)=1:2 and .^(n)C(r+1): ^(n)C(r+2)=2:3, find ...

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  2. If "^(n-1)Cr : ^nCr : ^(n+1)Cr= 6:9: 13, find n and r.

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  3. If m= "^nC2, prove that "^m C2 =3xx ^(n+1)C4.

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  4. Prove that "^nCr+2 ^(n)C(r-1)+ ^(n)C(r-2) = ^(n+2)Cr.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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