Combinatorics

A youth council in Kigali has $9$ members. In how many ways can a working group of $4$ members be chosen? (All members of the group have the same role.)

Find all whole numbers $k$ with $\binom{10}{k} = 120$. List them separated by commas.

How many five-card hands from a standard pack of $52$ cards contain at least one ace? (There are $\binom{52}{5} = 2\,598\,960$ hands in all, and $\binom{48}{5} = 1\,712\,304$.)

What is the coefficient of $x^3$ in the expansion of $(2x + 1)^5$?

In the proof of Pascal's rule $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$, the $k$-person committees chosen from $n$ people are split by whether they contain Amina. Why are there exactly $\binom{n-1}{k-1}$ committees containing Amina?

In a class of $40$ students in Accra, $22$ play football, $15$ play basketball and $8$ play both. How many students play neither sport?

What is the smallest number $m$ such that any $m$ different whole numbers chosen from $1, 2, \dots, 100$ must include two whose difference is divisible by $7$?

Here s is a finite set of objects, t a finite set of boxes and f assigns a box to each object. Which Lean statement correctly formalises the pigeonhole principle?

Write derangements(n) returning the number of ways to arrange $n$ hats so that nobody gets their own hat, using the inclusion–exclusion formula $D_n = \sum_{k=0}^{n} (-1)^k \dfrac{n!}{k!}$. Use exact integer arithmetic (math.factorial and //). Then check: does $D_n / n!$ approach $1/e \approx 0.3679$?