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Prove that (n !)^2<n^nn !<(2n)! for all ...

Prove that `(n !)^2

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We have
`(n!)^(2)=(n!)xx(n!)=(n!)xx(1xx2xx3xx...xxn)`.
`le (n!)xxn^(n) " "[because r le n " for each "r=1,2,3,...,n].`
`therefore (n!)^(2) le (n!)xx n^(n) " " `...(i)
Again, we have
`(2n)! =(1xx2xx3xx... xxn)xx (n+1)xx(n+2)xx...xx2n`
`gt (n!)xxn^(n). " "[because (n+r)gt n" for each " r=1,2,...,n].`
Thus, `(n!)xxn^(n) lt (2n)!." " `...(ii)
Hence, from (i) and (ii), we have
`(n!)^(2) le (n!) xx n^(n) lt (2n)!.`
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RS AGGARWAL-PERMUTATIONS-EXERCISE 8H (Very-Short-Answer Questions)
  1. Prove that (n !)^2<n^nn !<(2n)! for all positive integers ndot

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  2. If (n+1)! =12 xx [(n-1)!], find the value of n.

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  3. If (1)/(4!) + (1)/(5!)=(x)/(6!), find the value of x.

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  4. How many 3-digit numbers are there with no digit repeated?

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  5. How many 3-digit numbers above 600 can be formed by using the digits 2...

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  6. How many numbers divisible by 5 and lying between 4000 and 5000 can b...

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  7. In how many ways can the letters of the word 'CHEESE' be arranged?

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

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  9. How many different words can be formed by using all the letters of the...

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  10. How many permutations of the letters of the word 'APPLE' are there?

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  11. How many words can be formed by the letters of the word 'SUNDAY'?

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  12. In how many ways can 4 letters be posted in 5 letter boxes?

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  13. In how many ways can 4 women draw water from 4 taps, if no tap remains...

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  14. How many 5-digit numbers can be formed by using the digits 0, 1 and 2?

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  15. In how many ways can 5 boys and 3 girls be seated in a row so that eac...

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  16. A child has plastic toys bearing the digits 4, 4 and 5. How many 3-dig...

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  17. In how may ways can the letters of the word 'PENCIL' be arranged so th...

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