Q.In how many ways can a committee of 4 be selected from 8 people, and then a chairperson be chosen from among the 4 selected members?

[!TLDR] Select the committee first, then select 1 chairperson from within it, and multiply. [!ANSWER] $^{8}C_{4}\times4 = 70\times4 = 280$ ways.

First select the 4-member committee from 8 people: $^{8}C_{4} = \dfrac{8\times7\times6\times5}{4!} = \dfrac{1680}{24} = 70$ ways. Then, from within that committee of 4, choose 1 chairperson: 4 ways. By the Fundamental Principle of Counting: $70 \times 4 = 280$. (Cross-check via a different split: choose the chairperson first from all 8 people, 8 ways, then choose the remaining 3 committee members from the other 7: $^{7}C_{3}=35$, giving $8 \times 35 = 280$ — the same answer, confirming the relationship between selecting and arranging a distinguished role.) [!ANSWER] $280$ ways.

Select the committee via a combination ($^{8}C_{4}$), then select the chairperson from within the committee (4 ways), and multiply the two stages; optionally cross-check with the chairperson-first ordering.

A common mistake is treating this as a straightforward permutation ($^{8}P_{4}$) without realising the committee members other than the chair are unordered among themselves — only the chair role is distinguished.