Problem 21 · 2011 Math Kangaroo
Stretch
Counting & Probability
careful-countingcasework
A cat had 7 kittens. The kittens had the colours white, black, ginger, black-white, ginger-white, ginger-black, and ginger-black-white. In how many ways can you choose 4 cats so that each time two of them have a colour in common?
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Answer: C — 4
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Hint 1 of 2
Each kitten is a set of colours; every chosen pair must share a colour.
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Hint 2 of 2
Look for groups of four whose colour-sets pairwise overlap.
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Approach: count pairwise-intersecting families of four
- List the colour-sets; any two chosen kittens must share at least one colour.
- Three families work by sharing one fixed colour (all whites, all blacks, or all gingers — four kittens each).
- A fourth family is the three two-colour kittens plus the all-three kitten, which also pairwise overlap.
- That makes 4 valid ways to choose the four kittens.
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