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Question: Find the number of permutations of \[n\] distinct things taken \[r\] together, in which 3 particular...

Find the number of permutations of nn distinct things taken rr together, in which 3 particular things must occur together.
A) n−3Cr−3(r−2)!3!{}^{n - 3}{C_{r - 3}}\left( {r - 2} \right)!3!
B) n−3Cr−3(r−3)!3!{}^{n - 3}{C_{r - 3}}\left( {r - 3} \right)!3!
C) n−3Cr−3(r−2)!{}^{n - 3}{C_{r - 3}}\left( {r - 2} \right)!
D) None of these

Explanation

Solution

Here we will first find the possible number of arrangements when the three things come together. Then we will find the number of ways in which all the three objects can be arranged. We will then find the number of ways in which all the (r−3+1)\left( {r - 3 + 1} \right) objects can be arranged. Then we will combine all the ways to find the required number of permutations.

Complete Step by Step Solution:
It is given that 3 particular things must occur together which means the total number of objects that become n−3n - 3 taken things becomes r−3r - 3.
So the number of arrangement when the three things come together is =n−3Cr−3 = {}^{n - 3}{C_{r - 3}}
Now we will find the number of ways in which all the three objects can be arranged. Therefore, we get
Arrangement of 3 particular things which must occur together is =3! = 3!
Now we will find the number of ways in which all the (r−3+1)\left( {r - 3 + 1} \right) objects can be arranged. Therefore, we get
Number of arrangements of the (r−3+1)\left( {r - 3 + 1} \right) objects is =(r−3+1)!=(r−2)! = \left( {r - 3 + 1} \right)! = \left( {r - 2} \right)!
Now we will write the permutation of nn distinct things taken rr together, in which 3 particular things must occur together, we get
Total possible arrangements is =n−3Cr−3(r−2)!3! = {}^{n - 3}{C_{r - 3}}\left( {r - 2} \right)!3!
Hence the number of permutations of nn distinct things taken rr together, in which 3 particular things must occur together is n−3Cr−3(r−2)!3!{}^{n - 3}{C_{r - 3}}\left( {r - 2} \right)!3!.

Therefore, option A is the correct option.

Note:
Here we have to note that we need to apply the formula of permutation not the combination for the arrangement of the letters. As a permutation is defined as the different ways in which a collection of items can be arranged. For example: The different ways in which the numbers 1, 2 and 3 can be grouped together, taken all at a time, are123,132,213231,312,321123,{\rm{ }}132,{\rm{ }}213{\rm{ }}231,{\rm{ }}312,{\rm{ }}321.
Combinations may be defined as the various ways in which objects from a set may be selected. For example: The different selections possible from the numbers 1, 2, 3 taking 2 at a time, are 12,23and31{\rm{12, 23 and 31}}