Question

Samanta's parents took her to a theme park and told her that she could pick out three toys from the gift shop. If there are nine different toys in total, how many different selections of three toys could she make? Write out Pascal's Triangle from row 0 to row 9 and use it to solve the problem.

Answer 1

To find out how many different selections of three toys Samanta could make from the nine available toys, we can use combinations. The number of ways to choose \( k \) items from \( n \) items is given by the formula:

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

In this case, \( n = 9 \) (the total number of toys) and \( k = 3 \) (the number of toys Samanta is selecting). So we need to calculate \( \binom{9}{3} \).

Using the combination formula:

\[
\binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9!}{3!6!}
\]

Calculating the factorials, we get:

- \( 9! = 9 \times 8 \times 7 \times 6! \)
- \( 3! = 3 \times 2 \times 1 = 6 \)
- \( 6! \) cancels out in the numerator and the denominator.

Now substituting back into the equation:

\[
\binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = \frac{504}{6} = 84
\]

So, there are 84 different selections of three toys that Samanta could make.

Now, let's write out Pascal's Triangle from row 0 to row 9 to verify the result:

```
Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1
Row 5: 1 5 10 10 5 1
Row 6: 1 6 15 20 15 6 1
Row 7: 1 7 21 35 35 21 7 1
Row 8: 1 8 28 56 70 28 8 1
Row 9: 1 9 36 84 126 36 9 1
```

From Pascal's Triangle, the value in Row 9 and Column 3 is indeed 84, which corresponds to \( \binom{9}{3} = 84 \).

Thus, the final answer is:

\[
\boxed{84}
\]