387 views
2 votes
2 votes

There are 6 periods in each working day of a school. In how many ways can one organize 5 subjects such that each subject is allowed at least one period?

sol :

5 subjects can be arranged in 6 periods in 6P5 ways.

Any of the 5 subjects can be organized in the remaining period (5C1 ways).

Two subjects are alike in each of the arrangement. So we need to divide by 2! to avoid overcounting.

Total number of arrangements
= 6P5× 5C12!=1800= 6P5× 5C12!= 1800

1 Answer

Best answer
0 votes
0 votes

The division by 2! is to eliminate duplicate permutations which have been included in our result (by solving 6P5× 5C1 = 3600).

let me elaborate:

When we solve for the number of permutations by the above approach we are including 2 instances of each permutation. How? Let's look at it more intuitively:  suppose the the five subjects are S1,S2,S3,S4,S5 and we have to arrange these in 6 lectures so that each subject gets at least one lecture.

One of the permutations obtained from 6P5 will be :

S1,S2,S3,S4,S5,_  (here  _  denotes the last lecture for which we need to choose from any of S1,S2,S3,S4,S5). Suppose we fill this last lecture with S1 then we get a final permutation  of S1,S2,S3,S4,S5,S1

One other permutation results in the same final permutation as above. Suppose the initial permutation obtained from 6P5 is _,S2,S3,S4,S5,S1. Now we can again choose to fill the empty lecture ( that is _ ) with any of S1,S2,S3,S4,S5. Suppose we choose S1. The final result of this is again S1,S2,S3,S4,S5,S1

We see from above example that we get S1,S2,S3,S4,S5,S1 from two different permutations which are both included in our formula of 6P5 x 5C1. This is also true for all other permutations that we obtain. Therefore we have clearly shown that we are including two instances of each permutation, hence we need to divide this by 2 to get the actual result. This is the reason why we are dividing by 2!.

Hope this wasn't too complicated and feel free to ask for clarification via comment.

selected by

Related questions

0 votes
0 votes
0 answers
1
0 votes
0 votes
0 answers
2
`JEET asked Jan 16, 2019
149 views
A man borrows Rs. 12,500 at 20% compound interest. At the end of every year he pays Rs. 2000 as part repayment. How much does he still owe after three such installments?