Combinatorial examples

1. In the class you have 15 pupils. In how many ways can we select four for examination?
2. In how many ways can we select from seven-card cards (32 cards) any two cards?
3. In how many ways can we divide 12 pupils into two six-member teams?
4. In how many ways can a teacher select from 12 pupils three to carry books to the storeroom?
5. Five friends are saying goodbye. Each shakes hands with each other. How many handshakes were there?

Final Answer:

a =  1365
b =  496
c =  4096
d =  220
e =  10

Step-by-step explanation:

$$\begin{gathered} C_4(15)=\left(\genfrac{}{}{0ex}{}{15}{4}\right)=\frac{15!}{4!(15-4)!}=\frac{15\cdot 14\cdot 13\cdot 12}{4\cdot 3\cdot 2\cdot 1}=1365 \\ a=\left(\genfrac{}{}{0ex}{}{4}{15}\right)=1365 \end{gathered}$$

$$\begin{gathered} C_2(32)=\left(\genfrac{}{}{0ex}{}{32}{2}\right)=\frac{32!}{2!(32-2)!}=\frac{32\cdot 31}{2\cdot 1}=496 \\ b=\left(\genfrac{}{}{0ex}{}{2}{32}\right)=496 \end{gathered}$$

$c=2^{12}=4096$

$$\begin{gathered} C_3(12)=\left(\genfrac{}{}{0ex}{}{12}{3}\right)=\frac{12!}{3!(12-3)!}=\frac{12\cdot 11\cdot 10}{3\cdot 2\cdot 1}=220 \\ d=\left(\genfrac{}{}{0ex}{}{3}{12}\right)=220 \end{gathered}$$

$e=5\cdot 4/2=10$

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