Find the value of n such that \(i) nP5=42 nP3,n>4 \(ii) (n−1)P4nP4,n>4
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Step-by-Step Solution
Step 1: Define Permutation Formula
The permutation formula nPr represents the number of ways to arrange r items from a set of n distinct items. It is defined as n! divided by (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 on the left side and 42⋅nP3 on the right side.
Step 3: Solve for n in part (i)
Simplify the equation by canceling n! from both sides and expanding (n−3)! as (n-3)(n-4)(n-5)!. This leads to a quadratic equation. Solve the quadratic equation by factoring to find the possible values of n. Since n>4, we discard the negative solution.
Step 4: Set up and simplify the expression for part (ii)
Substitute the permutation formula for both nP4 and (n−1)P4. Then, simplify the complex fraction by multiplying by the reciprocal of the denominator. Expand n! as n⋅(n−1)! and (n−4)! as (n−4)⋅(n−5)! to cancel common terms.