Unit 2 test: Counting and Probability

Eight people stand in a row. In how many orders can they stand if three particular friends must stand together (in any order)?

A committee of $5$ is chosen from $7$ teachers and $6$ students. The committee must include at least $2$ teachers and at least $2$ students. How many committees are possible?

How many ordered quadruples $(a, b, c, d)$ of nonnegative integers satisfy $a + b + c + d = 15$ with $a \ge 2$ and $b \ge 1$?

How many four-digit positive integers have digits that never increase from left to right (such as $9530$, $7770$, or $4444$)?

How many integers from $1$ to $600$ are divisible by none of $4$, $6$, and $9$?

Five different tasks are assigned to four workers. Each task goes to one worker, and every worker must get at least one task. How many assignments are possible?

Three fair dice are rolled. What is the probability that all three show different numbers?

Three people are chosen at random from five married couples. What is the probability that no married couple is among the three chosen? Give your answer as a fraction.

A standard $52$-card deck ($26$ red, $26$ black) is shuffled and the cards are laid out in a row. What is the expected number of places where two adjacent cards are both red?

A box holds $3$ red balls and $2$ white balls. Balls are drawn one at a time, without replacement, until both white balls have been drawn. What is the expected number of draws?

Numbers $x$ and $y$ are chosen independently and uniformly from $[0, 1]$. What is the probability that the larger of the two is more than twice the smaller?

Three numbers $x$, $y$, $z$ are chosen independently and uniformly from $[0, 1]$. What is the probability that $x + y + z \lt 1$? Give your answer as a fraction.

In how many ways can a $1 \times 10$ strip be tiled using $1 \times 1$ squares and $1 \times 3$ tiles?

Alex and Blair take turns rolling a pair of fair dice, with Alex going first. Alex wins if she rolls a sum of $6$ on one of her turns; Blair wins if he rolls a sum of $7$ on one of his turns. They continue until someone wins. What is the probability that Alex wins?