Build and Use the Triangle

Draw Pascal's triangle by hand for 8 rows. Start with a 1 at the top. Each new row starts and ends with 1; every other number is the sum of the two directly above. Row 4 is 1, 4, 6, 4, 1. Row 5 is 1, 5, 10, 10, 5, 1.

Now use it. How many 5-card hands contain exactly 2 aces? Choose 2 aces from 4 (C(4,2) = 6) and 3 non-aces from 48 (C(48,3) = 17,296). Multiply: 6 × 17,296 = 103,776 hands out of 2,598,960 total, about 4%.

Try one more. Flip a coin 6 times. Chance of exactly 3 heads? C(6,3) = 20 out of 2⁶ = 64 arrangements: 20/64 ≈ 31%. The triangle tells you the odds of any "exactly k out of n" event without further calculation.