Discrete Math

Evaluate the binomial coefficient inom{7}{5}.

Evaluate the binomial coefficient when $n$ and $r$ are the same, for example, $\binom{9}{9}$.

Find the coefficient of the $x^3 y^5$ term in the expansion of $(2x-3y)^8$.

Find the coefficient of the $x^4y^{12}$ term in the expansion of $(5x^2-2y^3)^6$.

Determine the coefficient of the third term in the binomial expansion of $(2x + y)^5$.

Find the coefficient of $x^5$ in the binomial expansion of $(2x - 8)^8$.

Expand $(x + 2)^4$ using Pascal's triangle.

Expand $(2x-3y)^5$ using Pascal's triangle.

Given that the coefficient of $x^3$ is 3 times that of $x^2$ in the expansion $(2 + 3x)^n$, find the value of $n$.

Using Breadth First Search (BFS), find the shortest path between a start node and an end node in an unweighted graph.

Using the breadth-first search method, find all the nodes discoverable from the root node $A$ in a given graph.

Prove the recursive identity for binomial coefficients: inom{n}{k} = inom{n-1}{k} + inom{n-1}{k-1} given the restrictions $n extgreater 0$ and $0 extless k extless n$.

Prove that ${5 \choose 3} \times {3 \choose 2} = {5 \choose 2} \times {3 \choose 1}$.

Prove that the sum of binomial coefficients for a set of size $n$ equals $2^n$, i.e., sum_{k=0}^{n} {n \choose k} = 2^n.

Given a function named addup, which adds numbers up to a certain number depending on the argument passed, determine the runtime complexity of the function both for the scenario where it iterates with a for-loop and where it computes using the formula sum = $n \times (n + 1) / 2$.

Explain the different time complexities represented by Big O notation, such as $O(1)$, $O(n)$, $O(\log n)$, $O(n \log n)$, and $O(n^2)$, using examples like accessing an element in an array, binary search, looping through elements, and sorting operations.

What is the probability that you're an alcoholic given that you're a male?

There are 500 students in a certain School. 150 students are enrolled in an algebra course and 80 students are enrolled in a chemistry course. There are 30 students who are taking both algebra and chemistry. If a student is chosen at random, what is the probability that the student is taking algebra?

What is the probability that the student is taking chemistry given that the student is also taking algebra?

What is the probability that the student is taking algebra given that the student is also taking chemistry?