Pigeonhole Example Quiz: Socks In A Drawer Pigeonhole

1) A drawer has socks of 3 different colors. How many socks must you draw (without looking) to guarantee at least one matching pair?

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5

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Use the pigeonhole principle with c=3 colors and m=2. Threshold = (m−1)·c + 1 = 1·3 + 1 = 4. With 4 draws, a pair is guaranteed.

Explanation

Use the pigeonhole principle with c=3 colors and m=2. Threshold = (m−1)·c + 1 = 1·3 + 1 = 4. With 4 draws, a pair is guaranteed.

2) With c colors available, drawing c+1 socks guarantees a matching pair.

True

False

By the pigeonhole principle, placing c+1 items into c color boxes forces some color box to contain at least 2 socks.

Explanation

By the pigeonhole principle, placing c+1 items into c color boxes forces some color box to contain at least 2 socks.

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4) There are 5 colors. What is the minimum number of draws to guarantee 3 socks of the same color?

10

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11

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Use (m−1)·c + 1 with m=3, c=5 ⇒ 2·5 + 1 = 11.

Explanation

Use (m−1)·c + 1 with m=3, c=5 ⇒ 2·5 + 1 = 11.

5) Select all expressions that guarantee at least m socks of the same color when there are c colors.

(m−1)·c + 1

C + m

C·m

C + (m−1)

(m−1)·(c−1) + 1

Least‑favorable draws take at most m−1 from each color. After (m−1)·c draws you could still avoid m‑of‑a‑kind; the next draw, +1, forces it. Other expressions do not guarantee m‑of‑a‑kind.

Explanation

Least‑favorable draws take at most m−1 from each color. After (m−1)·c draws you could still avoid m‑of‑a‑kind; the next draw, +1, forces it. Other expressions do not guarantee m‑of‑a‑kind.

6) A drawer has exactly 2 red, 2 blue, and 2 green socks. Drawing 4 socks guarantees at least one matching pair.

True

False

Worst case: first 3 draws are one of each color (no pair). The 4th must match one of them, yielding a pair.

Explanation

Worst case: first 3 draws are one of each color (no pair). The 4th must match one of them, yielding a pair.

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8) A drawer has 4 red, 3 blue, and 5 green socks. Minimum draws to guarantee at least one pair?

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6

3

4

The guarantee depends on the number of colors, not the exact counts (as long as each color is present). With 3 colors, c+1 = 4 draws guarantee a pair.

Explanation

The guarantee depends on the number of colors, not the exact counts (as long as each color is present). With 3 colors, c+1 = 4 draws guarantee a pair.

9) Which scenarios force at least one matching pair by the pigeonhole principle?

8 draws from 7 colors

10 draws from 3 colors

5 draws from 5 colors

12 draws from 11 colors

4 draws from 3 colors

A: 8>7, B: 10>3, D: 12>11, E: 4>3 all guarantee a pair. C: 5=5 may have all different, so no guarantee.

Explanation

A: 8>7, B: 10>3, D: 12>11, E: 4>3 all guarantee a pair. C: 5=5 may have all different, so no guarantee.

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10) With 7 colors, drawing 8 socks guarantees at least one pair.

True

False

Threshold for a pair with c=7 is c+1 = 8, so the statement is true.

Explanation

Threshold for a pair with c=7 is c+1 = 8, so the statement is true.

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12) With 4 colors, what is the minimum draws to guarantee 3 socks of one color?

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Use (m−1)·c + 1 with m=3, c=4 ⇒ 2·4 + 1 = 9. Wait, compute carefully: 2·4 + 1 = 9. The correct option is C=9. Adjusting selection.

Explanation

Use (m−1)·c + 1 with m=3, c=4 ⇒ 2·4 + 1 = 9. Wait, compute carefully: 2·4 + 1 = 9. The correct option is C=9. Adjusting selection.

13) With 4 colors, what is the minimum draws to guarantee 3 socks of one color?

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Apply (m−1)·c + 1 with m=3, c=4 ⇒ 2·4 + 1 = 9.

Explanation

Apply (m−1)·c + 1 with m=3, c=4 ⇒ 2·4 + 1 = 9.

14) For c=4 colors, which thresholds are correct? Select all that apply.

Guarantee a pair: 5

Guarantee a pair: 4

Guarantee a triple: 9

Guarantee four‑of‑a‑kind: 13

Guarantee five‑of‑a‑kind: 17

Using (m−1)·4 + 1: m=2→5, m=3→9, m=4→13, m=5→17. Option B (4) is too small.

Explanation

Using (m−1)·4 + 1: m=2→5, m=3→9, m=4→13, m=5→17. Option B (4) is too small.

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15) A drawer has 1 red, 4 blue, and 4 green socks. Minimum draws to guarantee a pair?

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Worst case: draw red, blue, green (3 different). The 4th draw must match one of them, so 4 draws guarantee a pair.

Explanation

Worst case: draw red, blue, green (3 different). The 4th draw must match one of them, so 4 draws guarantee a pair.

16) If there are only two colors, drawing 3 socks guarantees a matching pair.

True

False

With c=2, c+1=3 ensures a pair.

Explanation

With c=2, c+1=3 ensures a pair.

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18) There are 6 colors. Minimum draws to guarantee 3 socks of the same color?

11

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13

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Use (m−1)·c + 1 with m=3, c=6 ⇒ 2·6 + 1 = 13.

Explanation

Use (m−1)·c + 1 with m=3, c=6 ⇒ 2·6 + 1 = 13.

19) Select all statements that are always true.

For c colors, (m−1)·c + 1 draws guarantee m of a color

For c colors, c+1 draws guarantee a pair

With c colors, drawing c socks always guarantees a pair

If a color is absent (has 0 socks), the (m−1)·c + 1 bound is still sufficient

If some colors have fewer than m socks, more than (m−1)·c + 1 draws may be needed

A and B are direct pigeonhole results. C is false (you could draw one of each). D is still sufficient (fewer effective colors tighten the bound). E is false; the sufficient bound never needs to increase.

Explanation

A and B are direct pigeonhole results. C is false (you could draw one of each). D is still sufficient (fewer effective colors tighten the bound). E is false; the sufficient bound never needs to increase.

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20) Drawing 2c socks from c colors always guarantees at least two different matching pairs.

True

False

False: you can avoid a second pair by repeatedly drawing the same color after the first pair (e.g., counts 4,1,1,…). No second pair is forced.

Explanation

False: you can avoid a second pair by repeatedly drawing the same color after the first pair (e.g., counts 4,1,1,…). No second pair is forced.

21) A drawer has 3 red, 3 blue, and 1 green sock. Minimum draws to guarantee a pair is ____.

Worst case: draw one of each color (3 draws with no pair). The 4th draw must match red or blue, guaranteeing a pair.

Explanation

Worst case: draw one of each color (3 draws with no pair). The 4th draw must match red or blue, guaranteeing a pair.

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Pigeonhole Example Quiz: Socks in a Drawer Pigeonhole