Exercise 6.2 — Question 11

Combinations: Core Formula

A combination counts the number of ways to choose $r$ objects from $n$ distinct objects without regard to order:

${}^n{C}_r=\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\frac{n!}{r!(n-r)!}$

where $0\le r\le n$.

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Question 11

Problem: Find the number of ways a committee of 5 members can be formed from 8 persons.

Solution:

We need to choose 5 persons from 8, where order does not matter (a committee has no ranked positions).

${}^8{C}_5=\frac{8!}{5!(8-5)!}=\frac{8!}{5!\cdot 3!}$

$=\frac{8\times 7\times 6\times 5!}{5!\times 3!}=\frac{8\times 7\times 6}{3\times 2\times 1}=\frac{336}{6}=56$

Answer: The committee can be formed in 56 ways.

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Key Properties of Combinations Used

Why divide by $r!$?

Permutations count ordered arrangements. Since a combination ignores order, we divide by $r!$ (the number of ways to arrange $r$ chosen objects among themselves).

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Real-World Application

Combinations appear whenever selection without order matters:

• Choosing a lottery ticket (e.g., pick 6 numbers from 49): ${}^{49}{C}_6=13,983,816$ possible tickets.
• Selecting a team from a squad.
• Choosing a subset of DNA bases for a sequence segment.