Unit 2 test: Counting and Probability

Fifteen identical pencils are handed out to four students so that each student gets at least two pencils. In how many ways can this be done?

Find the number of $4$-element subsets of $\{1, 2, 3, \dots, 20\}$ in which every two elements differ by at least $4$.

How many integers from $1$ to $999$ are divisible by none of $4$, $6$, and $9$?

The numbers $1$ through $6$ are arranged in a random order in positions $1$ through $6$. The probability that exactly one number is in the position matching its value is $\dfrac{m}{n}$ in lowest terms. Find $m + n$.

A circular spinner is divided into $8$ congruent sectors, numbered $1$ through $8$ in order around the circle. Each sector is painted red, white, or blue so that no two sectors sharing an edge have the same color. How many colorings are possible?

A fair coin is flipped $12$ times. The probability that the flips never contain four consecutive heads is $\dfrac{m}{n}$ in lowest terms. Find $m + n$.

Four fair six-sided dice are rolled. The expected number of distinct values showing is $\dfrac{m}{n}$ in lowest terms. Find $m + n$.

A random permutation $a_1, a_2, \dots, a_6$ of $1, 2, \dots, 6$ is chosen. Call position $i$ a record if $a_i$ is larger than every earlier term $a_1, \dots, a_{i-1}$ (position $1$ is always a record). The expected number of records is $\dfrac{m}{n}$ in lowest terms. Find $m + n$.

Ari and Bex take turns rolling a fair die, with Ari first. Ari wins as soon as Ari rolls a $1$ or a $2$; Bex wins as soon as Bex rolls a $6$. The probability that Ari wins is $\dfrac{m}{n}$ in lowest terms. Find $m + n$.

A fair coin is flipped until the sequence $HTHT$ appears for the first time. What is the expected number of flips?

A token starts at one vertex of a regular pentagon. Each second it moves to one of the two adjacent vertices, each with probability $\dfrac{1}{2}$. The probability that the token is at its starting vertex after exactly $7$ moves is $\dfrac{m}{n}$ in lowest terms. Find $m + n$.

Four fair six-sided dice are rolled. In how many of the $1296$ outcomes is the sum $15$?

How many subsets of $\{1, 2, 3, \dots, 10\}$ (including the empty set) have a sum that leaves remainder $2$ when divided by $5$?

Find $\displaystyle\sum_{k=0}^{5}\binom{6}{k}\binom{6}{k+1}$.

Find the remainder when $\displaystyle\sum_{k=1}^{10} k\binom{10}{k}^2$ is divided by $1000$.