Problem 26 · AMC 8 Stretch
Stretch
Counting & Probability
and-process-multiplyconsidering-extreme-cases
A tattoo shop offers 6 different designs. No one gets the same design twice. (a) Bob may get any number of them, even none. How many different sets of tattoos could Bob have? (b) Suppose Bob definitely got the eagle design. Now how many different sets are possible?
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Answer: (a) 64; (b) 32
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Hint 1 of 4
Since no design repeats, a set of tattoos is just a subset of the 6 designs — each design is in or out.
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Hint 2 of 4
(a) Any number, even none, means count ALL subsets: \(2^6\).
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Hint 3 of 4
(b) If the eagle is definitely in, it is no longer a choice. Only the other 5 designs are still in-or-out.
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Approach: Subsets; fix an element for part (b)
- Since no design is repeated, a set of tattoos is a subset of the 6 designs.
- (a) Any number, even none: each design is in or out (AND process), so \(2^6 = 64\).
- (b) The eagle is definitely in, so that design is fixed and only the other 5 are still free choices: \(2^5 = 32\).
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