To organize 5 subjects in 7 periods in a working day such that each subject appears at least once, we can use a combinatorial approach similar to the known problem of arranging 5 subjects in 6 periods. Step-by-step reasoning:
- Since there are 7 periods and only 5 subjects, at least two subjects will have to be repeated to fill all periods.
- We want to count the number of ways to arrange these subjects so that each subject appears at least once.
- The problem can be seen as arranging 7 slots with 5 distinct subjects, where two subjects appear twice and the other three appear once.
- The number of ways to choose which 2 subjects will be repeated twice is (52)=10\binom{5}{2}=10(25)=10.
- Once the subjects and their frequencies are fixed (two subjects twice, three subjects once), the total number of distinct arrangements is the multinomial coefficient:
7!2!×2!×1!×1!×1!=50404=1260\frac{7!}{2!\times 2!\times 1!\times 1!\times 1!}=\frac{5040}{4}=12602!×2!×1!×1!×1!7!=45040=1260
- Multiply by the number of ways to choose the repeated subjects:
10×1260=1260010\times 1260=1260010×1260=12600
- Finally, multiply by the number of ways to assign the 5 distinct subjects to the chosen roles (which is already accounted for by choosing the repeated subjects), so no further multiplication is needed.
Answer: There are 12,600 ways to organize 5 subjects in 7 periods such that each subject is assigned at least one period, with exactly two subjects repeated to fill the 7 periods. Note: This approach generalizes the logic from the known problem of 5 subjects in 6 periods, where one subject is repeated once, yielding 1,800 ways
. Here, with 7 periods, two subjects repeat once each, increasing the count accordingly.
