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Question: If \({}^n{P_7} = 42\left( {{}^n{P_5}} \right)\), then find \(n\)...

If nP7=42(nP5){}^n{P_7} = 42\left( {{}^n{P_5}} \right), then find nn

Explanation

Solution

Hint- Here, we will be using the general formula for permutations.

Given, nP7=42(nP5) →(1){}^n{P_7} = 42\left( {{}^n{P_5}} \right){\text{ }} \to {\text{(1)}}
Since, we know that the general formula for permutation is given by
Number of ways of arranging rr items out of nn items is nPr=n!(n−r)!{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}
Solving the given equation (1) using above formula, we get
n!(n−7)!=42[n!(n−5)!] ⇒n(n−1)(n−2)(n−3)(n−4)(n−5)(n−6)(n−7)!(n−7)!=42[n(n−1)(n−2)(n−3)(n−4)(n−5)!(n−5)!] ⇒n(n−1)(n−2)(n−3)(n−4)(n−5)(n−6)=42n(n−1)(n−2)(n−3)(n−4) ⇒(n−5)(n−6)=42⇒n2−11n+30=42⇒n2−11n−12=0 ⇒n2+n−12n−12=0⇒n(n+1)−12(n+1)=0⇒(n+1)(n−12)=0   \dfrac{{n!}}{{\left( {n - 7} \right)!}} = 42\left[ {\dfrac{{n!}}{{\left( {n - 5} \right)!}}} \right] \\\ \Rightarrow \dfrac{{n\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right)\left( {n - 4} \right)\left( {n - 5} \right)\left( {n - 6} \right)\left( {n - 7} \right)!}}{{\left( {n - 7} \right)!}} = 42\left[ {\dfrac{{n\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right)\left( {n - 4} \right)\left( {n - 5} \right)!}}{{\left( {n - 5} \right)!}}} \right] \\\ \Rightarrow n\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right)\left( {n - 4} \right)\left( {n - 5} \right)\left( {n - 6} \right) = 42n\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right)\left( {n - 4} \right) \\\ \Rightarrow \left( {n - 5} \right)\left( {n - 6} \right) = 42 \Rightarrow {n^2} - 11n + 30 = 42 \Rightarrow {n^2} - 11n - 12 = 0 \\\ \Rightarrow {n^2} + n - 12n - 12 = 0 \Rightarrow n\left( {n + 1} \right) - 12\left( {n + 1} \right) = 0 \Rightarrow \left( {n + 1} \right)\left( {n - 12} \right) = 0 \\\ \\\
Either n=−1n = - 1 or n=12n = 12
Since, the value of nn should always be positive so we will neglect n=−1n = - 1.
Therefore, the possible value of nn is 12.

Note- In these types of problems we have to check at the end that the values of nn we are getting are non-negative. If any value of nn comes out to be negative, then that value is not considered because that value is not feasible.