Question

maisie has 12 new books in her library at home. she sets a goal to read 8 books during the next 6 weeks. how many different groups of 8 books is it possible for maisie to read? enter your answer in the box.

Explanation:

Step1: Identify the problem type

This is a combination problem, where we need to find the number of ways to choose 8 books from 12, which is calculated using the combination formula \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n = 12 \) and \( k = 8 \).

Step2: Simplify the combination formula

We know that \( C(n,k)=C(n,n - k) \), so \( C(12,8)=C(12,12 - 8)=C(12,4) \).
The formula for \( C(12,4) \) is \( \frac{12!}{4!(12 - 4)!}=\frac{12!}{4!×8!} \).

Step3: Calculate the factorial values

\( 12! = 12×11×10×9×8! \), so substituting into the formula:
\( \frac{12×11×10×9×8!}{4!×8!} \). The \( 8! \) terms cancel out.
\( 4! = 4×3×2×1 = 24 \).
Now calculate \( \frac{12×11×10×9}{24} \).
\( 12×11×10×9 = 11880 \), and \( 11880\div24 = 495 \).

Answer:

495