Question 1

Suppose you have a set of 12 blocks: 6 are black, 4 are red, 1 is white, and 1 is blue. If the child puts the blocks in a line, how many distinct arrangements are possible?

Accepted Answer

$$\frac{12!}{6!4!1!1!}$$

Question 2

A committee of 7, consisting of 2 Republicans, 2 Democrats, and 3 Independents, is to be chosen from a group of 5 Republicans, 6 Democrats, and 4 Independents. How many committees are possible?

Accepted Answer

$$\binom{5}{2} \binom{6}{2} \binom{4}{3}$$

Question 3

How many distinct nonnegative integer-valued solutions are there to $$x_1$$ + $$x_2$$ + $$x_3$$ + $$x_4 = 20$?

Accepted Answer

$$\binom{23}{3}$$

Question 4

Expand $$(x + y)^3$ using the binomial theorem.

Accepted Answer

$$x^3$$ + $$3x^2y$$ + $$3xy^2$$ + $$y^3$$

Question 5

A police department consists of 10 officers. If 5 are patrolling, 2 are working full time at the station, and 3 are on reserve, how many different divisions into these 3 groups are possible?

Accepted Answer

$$\frac{10!}{5!2!3!}$$

Question 6

In a knockout tournament with 8 players, how many possible outcomes are there for the entire tournament (all rounds)?

Accepted Answer

$$8!$$

Question 7

How many subsets are there of a set consisting of $$n$$ elements?

Accepted Answer

$$2^n$$

Question 8

How many terms are there in the multinomial expansion of $$(x_1$ + x_2$ + \cdots + x_r)^n$$?

Accepted Answer

$$\binom{n + r - 1}{r - 1}$$

Question 9

If $$n$$ indistinguishable balls are distributed into $$r$$ distinguishable urns, how many different outcomes are possible?

Accepted Answer

$$\binom{n + r - 1}{r - 1}$$

Question 10

What is the probability that, when two dice are rolled, the sum of the upturned faces equals 7?

Accepted Answer

$$\frac{1}{6}$$

Question 11

If $$n$$ people are present in a room, what is the probability that no two of them share a birthday (assuming 365 days in a year)?

Accepted Answer

$$\frac{365 \cdot 364 \cdot 363 \cdots (365-n+1)}{365^n}$$

Question 12

What is the probability that, in a random ordering of $$N$$ hats among $$N$$ men, none of the men selects his own hat?

Accepted Answer

$$1 - 1 + \frac{1}{2!} - \frac{1}{3!} + \cdots + (-1)^N \frac{1}{N!}$$

Question 13

If $$E$$ and $$F$$ are events in a sample space, what is the probability that either $$E or$$ $$F$$ occurs?

Accepted Answer

$$P(E) + P(F) - P(EF)$$

Question 14

If $$E is an $$event, what is the probability that $$E$$ does not occur?

Accepted Answer

$$1 - P(E)$$

Question 15

Suppose a deck of 52 cards is shuffled and the cards are turned up one at a time until the first ace appears. What is the probability that the card following the first ace is the ace of spades?

Accepted Answer

$$\frac{1}{52}$$

Question 16

How many different 5-card poker hands are possible from a standard deck of 52 cards?

Accepted Answer

$$\binom{52}{5}$$

Question 17

What is the probability that a randomly dealt 5-card poker hand is a straight (not all of the same suit)?

Accepted Answer

$$\frac{10(4^5$ - 4)}{\binom{52}{5}}$$

Question 18

If $$n$$ balls are to be distributed into $$r$$ urns so that the $$ith $$urn contains at least $$m_i$$ balls ($$n \geq \sum_{i=1}^r m_i$$), how many ways are there to do this?

Accepted Answer

$$\binom{n - \sum_{i=1}^r m_i + r - 1}{r - 1}$$

Question 19

Which of the following is a valid combinatorial identity for binomial coefficients?

Accepted Answer

$$\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$$

Combinatorial Analysis and Axioms of Probability: Study Notes

스터디 가이드 - 연습 문제

학습 가이드 - 플래시카드