If there are 6 periods on each working day of a school, in how many ways

Question:

If there are 6 periods on each working day of a school, in how many ways can one arrange 5 subjects such that each subject is allowed at least one period?

Solution:

To find: number of ways of arranging 5 subjects in 6 periods.

Condition: at least 1 period for each subject.

5 subjects in 6 periods can be arranged in $P(6,5)$.

Remaining 1 period can be arranged in $\mathrm{P}(5,1)$

Formula:

Number of permutations of $n$ distinct objects among $r$ different places, where repetition is not allowed, is

$P(n, r)=n ! /(n-r) !$

Total arrangements $=\mathrm{P}(6,5) \times \mathrm{P}(5,1)=\frac{6 !}{(6-5) !} \times \frac{5 !}{(5-1) !}$

$=\frac{6 !}{1 !} \times \frac{5 !}{4 !}=720 \times 5=3600$

Total number of ways is 3600 ways.

 

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