Find the value of n such that \(i) nP5=42^nP_{5} =42 nP3,n>4^nP_{3}, n>4 \(ii) nP4(n−1)P4,n>4\frac{^nP_{4}}{^(n-1)P_{4}} , n>4

Answer: (i) n=10n=10, (ii) nn−4\frac{n}{n-4}

Step-by-step solution

Step 1: Define Permutation Formula

The permutation formula nPr^nP_r represents the number of ways to arrange rr items from a set of nn distinct items. It is defined as n!n! divided by (n−r)!(n-r)!.

Step 2: Set up the equation for part (i)

Substitute the permutation formula into the given equation for part (i). We have nP5^nP_5 on the left side and 42⋅nP342 \cdot ^nP_3 on the right side.

Step 3: Solve for n in part (i)

Simplify the equation by canceling n!n! from both sides and expanding (n−3)!(n-3)! as (n−3)(n−4)(n−5)!(n-3)(n-4)(n-5)!. This leads to a quadratic equation. Solve the quadratic equation by factoring to find the possible values of nn. Since n>4n>4, we discard the negative solution.

Step 4: Set up and simplify the expression for part (ii)

Substitute the permutation formula for both nP4^nP_4 and (n−1)P4^(n-1)P_4. Then, simplify the complex fraction by multiplying by the reciprocal of the denominator. Expand n!n! as n⋅(n−1)!n \cdot (n-1)! and (n−4)!(n-4)! as (n−4)⋅(n−5)!(n-4) \cdot (n-5)! to cancel common terms.

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