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Question: If we are given the ratio of permutation as \({}^{n}{{P}_{4}}:{}^{n}{{P}_{5}}=1:2\), then \(n\) is: ...

If we are given the ratio of permutation as nP4:nP5=1:2{}^{n}{{P}_{4}}:{}^{n}{{P}_{5}}=1:2, then nn is:
1. 4
2. 5
3. 6
4. 7

Explanation

Solution

For solving this question you should know about the general formula of permutations. Here in this question we will set the given terms according to the formula of permutation and then we will compare that with the given ratio and find the value of nn. The formula of permutations is given by nPr=n!(n−r)!^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}.

Complete step-by-step solution:
According to the question given to us we are asked to find the value of nn if nP4:nP5=1:2{}^{n}{{P}_{4}}:{}^{n}{{P}_{5}}=1:2. So, if we take the expression given to us, then,
nP4:nP5=1:2………(i){}^{n}{{P}_{4}}:{}^{n}{{P}_{5}}=1:2\ldots \ldots \ldots \left( i \right)
Since we know that the general formula for permutations is given by, that is the number of ways of arranging rr items out of nn items is denoted as nPr=n!(n−r)!^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}.
Solving the given equation (i) using the above formula, we will get as follows,
n!(n−4)!n!(n−5)!=12 ⇒(n−5)!(n−4)!=12 \begin{aligned} & \dfrac{\dfrac{n!}{\left( n-4 \right)!}}{\dfrac{n!}{\left( n-5 \right)!}}=\dfrac{1}{2} \\\ & \Rightarrow \dfrac{\left( n-5 \right)!}{\left( n-4 \right)!}=\dfrac{1}{2} \\\ \end{aligned}
Simplifying further we will get as follows,
n!(n−4)(n−5)!:n!(n−5)!=12 ⇒n!(n−4)(n−5)!×(n−5)!n!=12 ⇒1(n−4)=12 ⇒n−4=2 ⇒n=6 \begin{aligned} & \dfrac{n!}{\left( n-4 \right)\left( n-5 \right)!}:\dfrac{n!}{\left( n-5 \right)!}=\dfrac{1}{2} \\\ & \Rightarrow \dfrac{n!}{\left( n-4 \right)\left( n-5 \right)!}\times \dfrac{\left( n-5 \right)!}{n!}=\dfrac{1}{2} \\\ & \Rightarrow \dfrac{1}{\left( n-4 \right)}=\dfrac{1}{2} \\\ & \Rightarrow n-4=2 \\\ & \Rightarrow n=6 \\\ \end{aligned}
Hence, we get the value of nn as 6 and so the correct answer is option 3.

Note: While solving such types of questions, we have to always check at the end if the final value that we are getting for nn is a non-negative value or not. If by any chance you get the value of nn as a negative one, then that value is not considered because that value is not a feasible value. So, always remember to check for the same in such questions.