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
How many valid (properly matched) sequences can be formed using 4 pairs of parentheses?
Answer: 14
The count is the Catalan number C4 = C(8,4)/5 = 14.
A and B take turns flipping a fair coin, A first, and the first to flip heads wins. What is the probability that A wins?
Answer: 2/3
P = 1/2 + (1/4)P gives P = 2/3.
How many positive divisors does 360 have?
Answer: 24
360 = 2^3·3^2·5, so it has (3+1)(2+1)(1+1) = 24 divisors.
A family has two children, and at least one is a boy. Assuming each child is equally likely to be a boy or girl, what is the probability both are boys?
Answer: 1/3
Of the equally likely outcomes BB, BG, GB, only BB works, giving 1/3.
How many rectangles of any size are in a 4×4 grid of unit squares?
Answer: 100
Choose 2 of 5 vertical and 2 of 5 horizontal lines: C(5,2)^2 = 100.
In how many ways can 5 people sit in a row if persons A and B must not sit next to each other?
Answer: 72
Total 5! = 120 minus 4!·2 = 48 adjacent arrangements gives 72.
Two numbers are chosen independently and uniformly from [0, 1]. What is the probability that they differ by less than 1/2?
Answer: 3/4
The complement is two corner triangles with total area 1/4, so the answer is 3/4.
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