Question

Find n if

Answer 1

The given equation is P n−1 3 : P n 4 =1:9.

The ratio means the fraction of the terms, so the given equation is written as,

P n−1 3 P n 4 = 1 9

The formula to calculate the permutation is,

P n r = n! ( n−r )!

Where,n is the number of objects taken r at a time.

Use formula of permutation to find the value of n.

( n−1 )! ( n−1−3 )! n! ( n−4 )! = 1 9 ( n−1 )! ( n−4 )! n! ( n−4 )! = 1 9

Cancel the common factors,

( n−1 )! n! = 1 9

Cancel the common factors by factorizing the bigger term to the factorial.

The formula to calculate the factors of a factorial in terms of factorial itself is,

n!=n( n−1 )! n!=n( n−1 )( n−2 )!                 [ n≥2 ] ⋮

Further simplify,

( n−1 )! n( n−1 )! = 1 9 1 n = 1 9

Reverse the above equation to calculate the value of n.

n=9

Thus, the value of n is 9.