Byzu's practice book

Question

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?

A. 3200

B. 3600

C. 1800

D. None of these

Comments

This is a simple DODB problem

We have 5 Subjects(Boxes) and we need to distribute 6 periods(chocolates) such that each Subject get atleast one period. Hence any one subject will get 2 periods.

Choose the subject which will be having 2 periods and allot it, allot rest subjects 1 period each

Total number of ways = $_{1}^{5}\textrm{C}*_{2}^{6}\textrm{C}*_{1}^{4}\textrm{C}*_{1}^{3}\textrm{C}*_{1}^{2}\textrm{C}*_{1}^{1}\textrm{C}$ = $1800$

Option C

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Another way to look at this problem, we have to find number of onto functions from set of periods to set of subjects.

Number of onto functions = Total functions - $($functions where $1$ subject doesn't have any preimage - functions where $2$ subjects don't have any preimage + functions where $3$ subjects don't have any preimage - functions where $4$ subjects don't have any preimage$)$

 $= 5^6 - \{ {5 \choose 1} * 4^5 - {5 \choose 2} * 3^5 + {5 \choose 3} * 2^5 - {5 \choose 4} * 1^5\} = 1800$