In the row of Pascal's triangle that starts with 1 and then 12, what is the fourth number?
The row that starts 1, 12 is the row $\binom{12}{0}, \binom{12}{1}, \binom{12}{2},\binom{12}{3}$, so the fourth number in this row is $\binom{12}{3} = \frac{12\cdot 11\cdot 10}{3\cdot 2 \cdot 1} = \boxed{220}$.
220
Level 3
Counting & Probability
Problem: In the row of Pascal's triangle that starts with 1 and then 12, what is the fourth number?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the fourth number in the row of Pascal's triangle that starts with 1 and then 12, we first identify which row this is. The second number in the row corresponds to the binomial coefficient C(n, 1), which equals n. Since the second number is 12, we have n = 12. Therefore, we are looking at the 12th row of Pascal's triangle.

The numbers in the nth row of Pascal's triangle are given by the binomial coefficients C(n, k) for k = 0, 1, 2, ..., n. The fourth number corresponds to k = 3. Thus, we need to calculate C(12, 3).

The formula for the binomial coefficient C(n, k) is:

C(n, k) = n! / (k! * (n - k)!)

Substituting n = 12 and k = 3 into the formula gives:

C(12, 3) = 12! / (3! * (12 - 3)!) = 12! / (3! * 9!)

We can simplify this expression. The factorial 12! can be expressed as:

12! = 12 × 11 × 10 × 9!

Thus, we can cancel the 9! in the numerator and denominator:

C(12, 3) = (12 × 11 × 10) / (3!)

Next, we calculate 3!:

3! = 3 × 2 × 1 = 6

Now we substitute this back into our expression:

C(12, 3) = (12 × 11 × 10) / 6

Calculating the numerator:

12 × 11 = 132
132 × 10 = 1320

Now we divide by 6:

1320 / 6 = 220

Thus, the fourth number in the row of Pascal's triangle that starts with 1 and then 12 is 220.<|END_OF_LATENT|> The final answer is $220$.