Combinatorics & Probability Flashcards
7 cards from real AMC 12 practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.
Read the first 7 Combinatorics & Probability flashcards as text
What is the coefficient of x^3 in the expansion of (1 + 2x)^5?
Answer: 80
C(5,3)·2^3 = 10·8 = 80.
A drawer holds socks in 4 colors. What is the minimum number drawn (without looking) to guarantee a matching pair?
Answer: 5
By the pigeonhole principle, 4 socks could all differ, so 5 guarantees a pair.
A point (x, y) is chosen uniformly at random in the unit square. What is the probability that x + y < 1/2?
Answer: 1/8
The region is a right triangle with legs 1/2, so its area is 1/8.
In how many ways can 7 identical balls be placed into 3 distinct boxes so that each box gets at least one ball?
Answer: 15
Place one ball per box first, then distribute 4 freely: C(6,2) = 15.
What is the expected number of fixed points in a uniformly random permutation of 8 elements?
Answer: 1
By linearity of expectation, each of 8 positions is fixed with probability 1/8, giving 1.
A fair die is rolled 4 times. What is the probability of rolling at least one 6?
Answer: 671/1296
1 − (5/6)^4 = 1 − 625/1296 = 671/1296.
How many 4-digit positive integers have strictly increasing digits from left to right?
Answer: 126
Zero can't appear, so choose 4 digits from 1–9 in one order: C(9,4) = 126.
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