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Prove that: ((2n)!)/(n !)={1. 3. 5 (2n-1...

Prove that: `((2n)!)/(n !)={1. 3. 5 (2n-1)}2^ndot`

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To prove that \(\frac{(2n)!}{n!} = (1 \cdot 3 \cdot 5 \cdots (2n-1)) \cdot 2^n\), we will start with the left-hand side and manipulate it step by step. ### Step 1: Write the left-hand side We start with the left-hand side of the equation: \[ \frac{(2n)!}{n!} \] ...
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RD SHARMA-PERMUTATIONS-Solved Examples And Exercises
  1. Prove that (n !)^2 < n^n n! < (2n)!, for all positive integers n.

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  2. Prove that (n !+1) is not divisible by any natural number between 2 a...

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  3. Prove that: ((2n)!)/(n !)={1. 3. 5 (2n-1)}2^ndot

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  4. If (n !)/(2!(n-2)!) and (n !)/(4!(n-4)!) are in the ratio 2:1 , find t...

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  5. Find n , if (n+2)! =2550xxn ! (ii) (n+1)! =12xx(n-1)!

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  6. If 1/(9!)+1/(10 !)=x/(11 !) , find xdot

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  7. If ((2n)!)/(3!(2n-3)!)a n d(n !)/(2!(n-2)!) are in the ratio 44 :3, fi...

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  8. How many permutations of the letters of the word MADHUBANI do not b...

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  9. Find the probability that in a random arrangement of the letters of th...

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  10. How many numbers can be formed with the digits 1, 2, 3, 4, 3, 2, 1 so ...

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  11. If the letters of the word ‘MOTHER’ are written in all possible orders...

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  12. In how many ways can the letters of the word PERMUTATIONS be arrang...

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  13. In how many ways can the letters of the word PERMUTATIONS be arrange...

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  14. (i) How many different words can be formed with letters of the word ‘S...

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  15. How many words can be formed using the letter A thrice, the letter B ...

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  16. How many 4 letters words can be formed using the letters of the words ...

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  17. m men and n women are to be seated in a row so that no two ...

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  18. A gentle man want to invite six friends. In how many ways and he se...

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  19. How many three-digit numbers more than 600 can be formed by using the ...

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  20. How many numbers between 3000 and 4000 can be formed form the digits 3...

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