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Prove that .^(15)C(3) + 2 xx .^(15)C(4)...

Prove that `.^(15)C_(3) + 2 xx .^(15)C_(4) + .^(15)C_(5) = .^(17)C_(5)`

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The correct Answer is:
`.^(40)C_(5)`
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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 ) .

The sum 2 xx .^(40)C_(2) + 6 xx .^(40)C_(3) + 12 xx .^(40)C_(4) + 20 xx .^(40)C_(5) + "…." + 1560 xx .^(40)C_(40) is divisible by

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

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SURA PUBLICATION-COMBINATORICS AND MATHEMATICAL INDUCTION -EXERCISE 4.3
  1. If .^(15)C(2r-1) = .^(15)C(2r + 4) " find " r

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  2. If .^(n)P(r) = 720 . " If " .^(n)C(r) = 120 " find " n , r

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  3. Prove that .^(15)C(3) + 2 xx .^(15)C(4) + .^(15)C(5) = .^(17)C(5)

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  4. Prove that .^(35)C(5) + sum(r=0)^(4^((39-r))) C(4) = .^(40)C(5)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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