Question

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

where Symbol have the usual meanings.

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Answer 1

Answer:

n = 19

Step-by-step explanation:

Given :-

C(n+2,8) : P(n-2,4) = 57:16

To find :-

Find the value of n ?

Solution :-

Given that

C(n+2,8) : P(n-2,4) = 57:16

C(n+2,8) /P(n-2,4) = 57/16 -----(1)

We know that

n C r = n! / (r! × (n-r)!

Now,

C(n+2,8) = (n+2) C 8

Where , n = n+2 and r = 8

=> (n+2)! / ( 8! × (n+2-8)!)

=> (n+2)! / ( 8!×(n-6)!)

=>(n+2)(n+1)(n)(n-1)(n-2)(n-3)(!/(8!×(n-6)!)

=(n+2)(n+1)(n)(n-1)(n-2)(n-3)(n-4)(n-5)/8!

-------(2)

and

We know that

nPr = n! / (n-r)!

P(n-2,4) = (n-2) P 4

Where , n = n-2 and r = 4

=> (n-2)! /(n-2-4)!

=> (n-2)!/(n-6)!

=> (n-2)(n-3)(n-4)(n-5)(n-6)!/(n-6)!

=> (n-2)(n-3)(n-4)(n-5) -------------(3)

On dividing (1) by (2)

=>[(n+2)(n+1)(n)(n-1)(n-2)(n-3)(n-4)(n-5)/8!]/[(n-2)(n-3)(n-4)(n-5)]

=> (n+2)(n+1)(n)(n-1)/8!

According to the given problem

=> (n+2)(n+1)(n)(n-1)/8! = 57/16

=> (n+2)(n+1)(n)(n-1)/(8×7×6×5×4×3×2×1) = 57/16

=> (n+2)(n+1)(n)(n-1)/40320 = 57/16

=> (n+2)(n+1)(n)(n-1)/2520 = 57

=> (n+2)(n+1)(n)(n-1) = 57×2520

=> (n+2)(n+1)(n)(n-1) = 143640

143640 can be written as 21×20×19×18

=> (n+2)(n+1)(n)(n-1) = 21×20×19×18

=> (n-1)(n)(n+1)(n+2) = 18×19×20×21

=> (n-1)(n)(n+1)(n+2) = (19-1)(19)(19+1)(19+2)

On comparing both sides then

=> n = 19

Therefore, n = 19

Answer:-

The value of n for the given problem is 19

Used formulae:-

→ n C r = n! / (r! × (n-r)!

→ nPr = n! / (n-r)!

→ n ! = (n)(n-1)(n-2)(n-3)...(3)(2)(1)

→ a:b can be written as a/b