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If the number of permutations of n diffe...

If the number of permutations of n different things taken r at time be denoted by `.^(n)P_(r)` , show that , `(.^(n)P_(1))/(1!)+(.^(n)P_(2))/(2!)+(.^(n)P_(3))/(3!)+...+(.^(n)P_(n))/(n!)=2^(n)-1`

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Let .^(n)P_(r) denote the number of permutations of n different things taken r at a time . Then , prove that 1+1.^(1)P_(1)+2.^(2)P_(2)+3.^(3)P_(3)+...+n.^(n)P_(n)=.^(n+1)P_(n+1) .

If (.^(n)P_(r-1))/(a)=(.^(n)P_(r))/(b)=(.^(n)P_(r+1))/(c) prove that , b^(2)=a(b+c)

The number of combinations of n different things taken r at a time in which p particular things always occur is -

If ""(n)C_(r) denotes the number of combinations of n different things taken r at a time, then the vlaue of ""^(n)C_(r+1)+""^(n)C_(r-1) + 2. ""^(n) C_(r) is-

Show that , .^(n)P_(r)=n.^(n-1)P_(r-1)=(n-r+1).^(n)P_(r-1) .

Show that , (.^(n)C_(r)+^(n)C_(r-1))/(.^(n)C_(r-1)+^(n)C_(r-2))=(.^(n+1)p_(r))/(r.^(n+1)p_(r-1))

Show that the numbers of permutations of n different things taken all at a time in which m particular things are never together is (n-m)!(n-m+1)! .

Show that the total number of permutations of n diferent things not more than r things at a time (repetition being allowed) is (n(n^(r)-1))/(n-1) .

Prove that .^(n)P_(r)=.^(n-1)P_(r)+r.^(n-1)P_(r-1) .

If .^(n)P_(5)=20.^(n)P_(3) , find n.

CHHAYA PUBLICATION-PERMUTATION AND COMBINATION -EXERCISE 7B ( Short Answer Type Questions )
  1. If .^(n)C(r-1)=36,.^(n)C(r)=84and.^(n)C(r+1)=126 find n and r .

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  2. If (.^(n)C(r-1))/(a)=(.^(n)C(r))/(b)=(.^(n)C(r+1))/(c) n=(ab+2ac...

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  3. If the number of permutations of n different things taken r at time be...

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  4. For n in N , Prove that (n+1)[n!n+(n-1)!(2n-1)+(n-2)!(n-1)]=(n+2)!

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  5. Evaluate : .^(20)C(5)+sum(j=2)^(5).^(25-j)C(4)

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  6. If .^(n)C(1),^(n)C(2)and^(n)C(3) are in A.P ., find n .

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  7. Solve : ((2x+1)!)/((x+2)!)xx((x-1)!)/((2x-1)!)=(3)/(5),( x in N N)

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  8. Prove that if ngt7 then .^(n-1)C(3)+.^(n-1)C(4)gt^(n)C(3)

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  9. How many different triangles can be formed by joining the angular poi...

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  10. How many words each consisting of five different letters can be formed...

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  11. find the number of different words that can be formed from 12 consonan...

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  12. A person has got 15 acquaintances of whom 10 are relatives . In how ma...

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  13. In how many different ways can 9 men be selected from 15 men so as to ...

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  14. In how many ways can a committee of a 3 ladies and 4 gentlemen be appo...

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  15. m men and n women are to be seated in a row that no two women sit tog...

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  16. Eight prizes are to be distributed by a lottery . The first participa...

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  17. In an election there are 7 candidates and 4 members are to be elected...

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  18. In a plane there are 10 points out of which no three are collinear exc...

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  19. If 20 straight lines be drawn in a plane , no two of them being parall...

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  20. If 10 parallel lines in a plane are intersected by a family of anothe...

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