One set, counted two ways

Mathematics · Discrete Mathematics · ages 18-19

Name ______________________ Date ____________

Split 4-person committees from 9 people on whether they include Ana. Which term counts committees that include her?
- C(8,4)
- C(9,3)
- C(8,3)
- C(9,4)
What is the binomial coefficient C(5, 2)? Type the number.
Answer: ______________
For integers n >= 0 and 0 <= r <= n, the binomial coefficient C(n,r) is a whole number.
Circle one: True False
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
There are C(8,3) = 56 three-person committees from 8 people. Of these, C(7,3) = 35 exclude Ana. How many include Ana?
Answer: ______________
When Pascal's rule splits pairs from 6 people by whether they include Ana, what are the two cases?
- Pairs from the first 3 plus pairs from the last 3
- Pairs including Ana plus pairs excluding her
- Pairs times singles, then divided by 2
Row 5 of Pascal's triangle reads 1, 5, 10, 10, 5, 1. What do its entries sum to? Type the number.
Answer: ______________
A proof splits 4-person committees from 9 people into those containing Ana and those containing Bo. Why is this not a valid Pascal partition?
- Some committees contain both; others contain neither.
- The two cases have different sizes, so they cannot be added.
- Every committee must contain Ana, so the Bo case is unnecessary.
- Fixing a person changes committees into ordered arrangements.
For any integer n >= 0, why does C(n,0) + ... + C(n,n) = 2^n? Choose the counting proof.
- Both count all subsets: by size on the left and by independent in-or-out decisions on the right.
- Both count only nonempty subsets: by size on the left and by independent in-or-out decisions on the right.
- Both count ordered lists: by length on the left and by independent first-or-last placements on the right.
Choose any 3 people from disjoint pools of 4 artists and 5 writers. Which expression counts every team once by its artist count?
- C(4,0) + C(5,3) + C(4,3) + C(5,0), adding pool counts
- Sum of C(4,k) × C(5,3-k) for k = 0,1,2,3
- C(4,1) × C(5,2), counting only one possible team type
- C(4,3) × C(5,3), choosing three people from each pool

Answer key

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

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C(8,3) · Ana fills one place. Choose the other three members from the remaining eight people: C(8,3).
10 · C(5,2) = (5 × 4) / (2 × 1) = 20 / 2 = 10.
True · On this integer domain, C(n,r) counts r-element subsets. A count of subsets is a whole number.
The r-person groups from n people · C(n, r) counts groups of size r drawn from n people.
21 · Including Ana and excluding Ana partition all committees. The missing count is 56-35 = 21, also C(7,2).
Pairs including Ana plus pairs excluding her · Pascal's rule splits the pairs by whether they include the fixed person.
32 · Row 5 sums to 2 to the power 5, which is 32.
Some committees contain both; others contain neither. · The cases overlap for committees with both people and omit committees with neither. A valid partition must count every committee exactly once.
Both count all subsets: by size on the left and by independent in-or-out decisions on the right. · Each subset has exactly one size, so the row sum counts all subsets once. Independent include-or-exclude decisions also produce each subset once, in 2^n ways. Thus the counts are equal.
Sum of C(4,k) × C(5,3-k) for k = 0,1,2,3 · For each artist count k, choose k artists and 3-k writers. Add the four disjoint case counts: 10 + 40 + 30 + 4 = 84.