Question

o lesson 5.5 example 5 – using the combinations formula

1. you are listening to music. you have time to hear only 3 songs from your playlist of 16 songs. how many combinations of 3 songs are possible?

Explanation:

Step1: Recall Combination Formula

The combination formula is \( C(n, r)=\frac{n!}{r!(n - r)!} \), where \( n = 16 \) (total songs) and \( r = 3 \) (songs to choose).

Step2: Substitute Values

Substitute \( n = 16 \) and \( r = 3 \) into the formula: \( C(16, 3)=\frac{16!}{3!(16 - 3)!}=\frac{16!}{3!×13!} \).

Step3: Simplify Factorials

Since \( n!=n×(n - 1)×\dots×(n - r+1)×(n - r)! \), \( 16! = 16×15×14×13! \). So the formula becomes \( \frac{16×15×14×13!}{3!×13!} \). The \( 13! \) terms cancel out.

Step4: Calculate Numerator and Denominator

\( 3! = 3×2×1 = 6 \), and \( 16×15×14 = 3360 \). Then \( \frac{3360}{6}=560 \).

Answer:

560