Binomial Coefficients and Combinatorial Identities · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

One set, counted two ways

Mathematics · Discrete Mathematics · ages 18-19
Name ______________________   Date ____________

  1. What is the binomial coefficient C(5,2)?

    • 10
    • 15
    • 20
    • 25

  2. What is the binomial coefficient C(5, 2)? Type the number.

    Answer: ______________

  3. A binomial coefficient always comes out a whole number.

    Circle one:   True   False

  4. What does the binomial coefficient C(n, r) count?

    • The r-person groups from n people
    • The n-person groups from r people
    • The sum of row n of Pascal's triangle

  5. How many 3-person committees can be chosen from 7 people? Give a number.

    Answer: ______________

  6. In the Pascal split of pairs from 6 people on Ana, what are the two piles?

    • Pairs from the first 3 plus pairs from the last 3
    • Pairs holding Ana plus pairs skipping her
    • Pairs times singles, then divided by 2

  7. Row 5 of Pascal's triangle reads 1, 5, 10, 10, 5, 1. What do its entries sum to? Type the number.

    Answer: ______________

  8. Use Pascal's rule to find C(6,2), given C(5,2) = 10 and C(5,1) = 5.

    • 15
    • 10
    • 20
    • 11

  9. Dan adds row 4 as 4 plus 6 plus 4, getting 14, and says the row rule fails. What is wrong?

    • He dropped both 1s; with them the sum is 16, or 2 to the power 4
    • The row rule only works for odd rows
    • Row 4 sums to 14, so the rule fails sometimes

  10. A team counts 2-person teams from 5 people (2 juniors, 3 seniors) by junior count: C(2,0)C(3,2) + C(2,1)C(3,1) + C(2,2)C(3,0). What is the total?

    • 10
    • 6
    • 12
    • 8
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Answer key

For grown-ups. Fold this page away before handing over the rest.

One set, counted two ways W1-mt_lHXFvudTh6-s1

  1. 10 · C(5,2) = 5 times 4 divided by 2 times 1 = 10.
  2. 10 · C(5, 2) is 5 times 4 divided by 2 times 1, which is 10.
  3. True · A count of groups must be whole, so the coefficient always is.
  4. The r-person groups from n people · C(n, r) counts groups of size r drawn from n people.
  5. 35 · C(7,3) = 7 times 6 times 5 divided by 6 = 35.
  6. Pairs holding Ana plus pairs skipping her · Pascal splits every pair on whether it holds the fixed person.
  7. 32 · Row 5 sums to 2 to the power 5, which is 32.
  8. 15 · Pascal's rule splits on one element: C(6,2) = C(5,2) plus C(5,1) = 10 plus 5 = 15.
  9. He dropped both 1s; with them the sum is 16, or 2 to the power 4 · Row 4 is 1, 4, 6, 4, 1, and sums to 16, matching the rule.
  10. 10 · Counting 2-person teams from 5 people two ways gives 1 times 3 plus 2 times 3 plus 1 times 1 = 10, which equals C(5,2).
Worksheet · LightMySky