Question

If \[{}^{12}{P_r} = 1320\] then find the value of \[r\].
A. \[5\]
B. \[4\]
C. \[3\]
D. \[2\]

Answer 1

Hint: In this question, we need to find the value of \[r\] if \[{}^{12}{P_r} = 1320\]. For this, we need to use the concept of permutation. Whenever the order of the arrangements concerns, a permutation is a mathematical technique used to find the variety of possible arrangements in a set.

Formula used: The formulas for permutation and the factorial of a number that are useful for solving the given question are given below. Suppose we have n things and we need to arrange r things out of n things in such way that \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\]
Also, the factorial of a number n (positive integer) is given by
\[n! = n \times \left( {n - 1} \right) \times .... \times 1\]

Complete step-by-step answer:
Consider \[{}^{12}{P_r} = 1320\] …. (1)
But we know that \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\] …. (2)
By comparing the equations (1) and (2), we get
\[n = 12\] and \[r = r\]
So, we get
\[{}^{12}{P_r} = \dfrac{{12!}}{{\left( {12 - r} \right)!}}\] …. From (2)
But \[{}^{12}{P_r} = 1320\]
Thus, we get
\[1320 = \dfrac{{12!}}{{\left( {12 - r} \right)!}}\]
Let us simplify this.
\[\left( {12 - r} \right)! = \dfrac{{12!}}{{1320}}\]
But \[n! = n \times \left( {n - 1} \right) \times .... \times 1\]
So, we get
\[\left( {12 - r} \right)! = \dfrac{{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{1320}}\]
By simplifying, we get
\[\left( {12 - r} \right)! = \dfrac{{{\text{479001600}}}}{{1320}}\]
\[\left( {12 - r} \right)! = {\text{362880}}\]
But \[{\text{362880}} = 9!\]
Thus, we get
\[\left( {12 - r} \right)! = {\text{9!}}\]
So, \[\left( {12 - r} \right) = {\text{9}}\]
By simplifying, we get
\[\left( {12 - 9} \right) = r\]
\[r = 3\]
Hence, the value of \[r\] is \[3\] if \[{}^{12}{P_r} = 1320\]
Therefore, the correct option is (C).

Additional information: We can say that a permutation is a specific arrangement of items. Set members or factors are organized in a sequence or linear order here. In simple terms, the arrangement is vital in permutations. To put it another way, the permutation is regarded as an ordered combination.

Note: Here, students generally make mistakes in calculating the factorial of a number. The main trick is after simplification, we get \[\left( {12 - r} \right)! = {\text{362880}}\]. But \[{\text{362880}} = 9!\]. This makes it easier to get the final result.