In how many ways can a committee of 4 people be chosen out of 8 people?

### Concept & Combination Basics Choosing a subgroup from a larger group without respect to arrangement order is a classic combination problem, solved using: $$ ^nC_r = \frac{n!}{r!(n-r)!} $$ ### Step-by-Step Solution 1. Identify total people available: $n = 8$. 2. Identify the number of people to choose: $r = 4$. 3. Apply the combination formula: $$ ^8C_4 = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} $$ 4. Simplify the expression: $4 \times 2$ in the denominator cancels the $8$ in the numerator, and $3$ cancels into $6$ (leaving 2). $$ 7 \times 2 \times 5 = 70 $$ ### Exam Strategy & Shortcut To calculate $^nC_r$ quickly, expand the numerator with $r$ descending terms and the denominator as $r!$. For $^8C_4$, just write out $\frac{8 \times 7 \times 6 \times 5}{24}$ and reduce it mentally. ### Common Pitfall Using permutation ($^8P_4 = 1680$) instead of combination. Committees do not imply any specific roles or order, so always use combinations. ### Final Answer Therefore, the correct answer is **70**.