Question

there are 40 volunteers for the research study on the Power Pill. Each subgroup of the study will contain 10 participants. Determine how many ways these participants can be selected and explain your method.

Answer 1

To form first group of 10 participants you have $C_{40}^{10}$ ways. After that only 30 volunteers left and to form second group of 10 participants you have $C_{30}^{10}$ ways. After forming second group 20 volunteers left and to form third group of 10 participants you have $C_{20}^{10}$ ways. At last, 10 volunteers left and to form fourth group of 10 participants you have $C_{10}^{10}$ ways.

Use product rule to count the total number of possible ways:

$C_{40}^{10}\cdot C_{30}^{10} \cdot C_{20}^{10}\cdot C_{10}^{10} =\dfrac{40!}{10!\cdot 30!} \cdot \dfrac{30!}{10!\cdot 20!} \cdot \dfrac{20!}{10!\cdot 10!} \cdot \dfrac{10!}{10!\cdot 0!} =\dfrac{40!}{(10!)^4}$