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Prove that .^(2n)C(n) = ( 2^(n) xx 1 x...

Prove that ` .^(2n)C_(n) = ( 2^(n) xx 1 xx 3 xx …(2n-1))/(n!)`

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Prove that .^(n)C_(0) +5 xx .^(n)C_(1) + 9 xx .^(n)C_(2) + "…." + (4n+1) xx .^(n)C_(n) = (2n+1) 2^(n) .

Prove that ((n + 1)/(2))^(n) gt n!

Prove that n! ( n +2) = n ! + ( n + 1) !

Prove that ((2n)!)/(n!) =2^(n) (1,3,5,……..(2n-1)) .

Prove that .^(n)C_(0) - ^(n)C_(1) + .^(n)C_(2)- ^(n)C_(3) + "…" + (-1)^(r) + .^(n)C_(r) + "…" = (-1)^(r ) xx .^(n-1)C_(r ) .

Prove that .^(n)C_(1) + 2 .^(n)C_(2) + 3 .^(n)C_(3) + "…." + n . ^(n)C_(n) = n 2^(n-1) .

Prove that (.^(n)C_(1))/(2) + (.^(n)C_(3))/(4) + (.^(n)C_(5))/(6) + "…." = (2^(n) - 1)/(n+1) .

Prove that ^n C_0 ^(2n)C_n- ^n C_1 ^(2n-2)C_n+ ^n C_2^(2n-4)C_n-=2^ndot

Prove that C_(0)^(2) + C_(1)^(2) + C_(2)^(2) +…+C_(n)^(2) = (2n!)/(n!)^(2)

SURA PUBLICATION-COMBINATORICS AND MATHEMATICAL INDUCTION -EXERCISE 4.3
  1. Prove that .^(35)C(5) + sum(r=0)^(4^((39-r))) C(4) = .^(40)C(5)

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  2. If .^(n+2) C(8) : ^((n-2)) P(4) = 57: 16 , find the value of n.

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  3. Prove that .^(2n)C(n) = ( 2^(n) xx 1 xx 3 xx …(2n-1))/(n!)

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  4. Prove that if 1 le r le n " then " n xx^((n-1))C(r-1)= (n-r+1).^(n)...

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  5. A Kabaddi coach has 14 players ready to play. How many different...

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  6. There are 15 persons in a party and if each 2 of then shakes hands w...

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  7. How many chords can be drawn through 20 points on a circle ?

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  8. In a parking lot one hundred one year old cars, are parked. Out of t...

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  9. How many ways can a team of 3 boys, 2 girls and 1 transgender be se...

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  10. Find the total number of subsets of a set with . (i) 4 elements (...

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  11. A trust has 25 members . (i) How many ways 3 officers can be sel...

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  12. How many ways a committee of six persons from 10 persons can be chosen...

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  13. How many different selections of 5 books can be made from 12 differe...

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  14. There are 5 teachers and 20 students . Out of them a committee of t...

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  15. In an examination a student has a answer 5 questions, out of 9 quest...

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  16. Determine the number of 5 card combinations out of a deck of 52 car...

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  17. Find the number of ways of forming a committee of 5 members out of...

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  18. A committee of 7 peoples has to be formed from 9 men and 4 women . In ...

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  19. 7 relatives of a man comprises 4 ladies and 3 gentlemen, his wife also...

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  20. A box contains two white balls, three black balls and four balls. In...

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