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Prove that: (i) r.^(n)C(r) =(n-r+1).^(...

Prove that:
(i) `r.^(n)C_(r) =(n-r+1).^(n)C_(r-1)`
(ii) `n.^(n-1)C_(r-1) = (n-r+1) .^(n)C_(r-1)`
(iii) `.^(n)C_(r)+ 2.^(n)C_(r-1) +^(n)C_(r-2) =^(n+2)C_(r)`
(iv) `.^(4n)C_(2n): .^(2n)C_(n) = (1.3.5...(4n-1))/({1.3.5..(2n-1)}^(2))`

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NAGEEN PRAKASHAN ENGLISH-PERMUTATION AND COMBINATION -Exercise F
  1. Evaluate the following: (i) .^(10)C(5) (ii) .^(12)C(8) (iii) .^(15...

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  2. Evaluate: .^(20)C(5)+^(20)C(4)+^(21)C(4)+^(22)C(4)

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  3. Prove that: (i) r.^(n)C(r) =(n-r+1).^(n)C(r-1) (ii) n.^(n-1)C(r-1)...

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  4. Find the value of n: (i) .^(n)C(10)=^(n)C(16) (ii) .^(15)C(n) =^(15...

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  5. If .^(n)C(10) = .^(n)C(15), then evaluate .^(27)C(n).

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  6. If .^(18)C(r) = .^(18)C(r+2), then evaluate .^(r)C(5).

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  7. If .^(n)C(5) = .^(n)C(7), then find .^(n)P(3)

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  8. If .^(16)C(r) = .^(16)C(r+6), then find .^(5)C(r).

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  9. Determine n if (i) ^2n C2:^n C2=12 :1 (ii) ^2n C3:^n C3=11 :1

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  10. Determine n if (i) .^(2n) C2: "^n C2=12 :1 (ii) .^(2n) C3: "^n C...

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  11. Determine n if (i) ^2n C2:^n C2=12 :1 (ii) ^2n C3:^n C3=11 :1

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  12. If .^(15)C(r): .^(15)C(r-1) = 1:5, then find r.

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  13. If .^(n-1)P3 :^(n+1)P3 = 5 : 12, find n.

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  14. If .^(n)P(r) = 720 and .^(n)C(r) = 120, then find r. .

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  15. If .^(n+1)C(r+1.): ^nCr: ^(n-1)C(r-1)=11:6:3 find the values of n and ...

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  16. If ""^(n)C(4),""^(n)C(5) and ""^(n)C(6) are in A.P. then the value of ...

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  17. If alpha=\ \ ^m C2,\ then find the value of \ ^(alpha)C2dot

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  18. In how many ways can a team of 11 players be selected from 14 players?

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  19. In how many ways 2 persons can be selected from 4 persons?

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  20. In how many ways can a person invites his 2 or more than 2 friends out...

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