Binomial Theorem

The number of terms in the expansion of $(x + y + z)^2$ is 6. How many terms are in the binomial expansion of $(x+y)^2$?

The value of $^{6}C_{3}$ is:

What is the second term in the expansion of $(1 - x)^{5}$?

Find the value of $^{9}C_{0} + ^{9}C_{9}$.

What is the sum of the coefficients in the expansion of $(4x - 3y)^n$?

If $A$ and $B$ are the coefficients of $x^p$ and $x^q$ in the expansion of $(1 + x)^{p+q}$, then:

The greatest binomial coefficient in the expansion of $(1 + x)^{10}$ is:

If $n$ is a positive integer, the coefficient of $x^n$ in the expansion of the product $(1 - x + x^2 - x^3 + \dots + (-1)^n x^n)(1 + x + x^2 + x^3 + \dots + x^n)$ is:

Find the coefficient of $x^5$ in the expansion of $(1+2x+3x^2+\dots+11x^{10})^2$.

The digit at the unit's place in the number $17^{100} + 11^{100} - 7^{100}$ is:

The diagram represents the coefficients of $(a+b)^n$ for $n=0$ to $n=3$. If we extend this pattern to $n=4$, what is the sum of all the coefficients in that specific row?

In the construction of Pascal's Triangle shown in the diagram, each internal entry is formed by the sum of the two entries directly above it. What is the value of the missing entry denoted by $x$ in the $5^{th}$ row (where the $0^{th}$ row is '1')?

What is the common ratio between the sum of elements of the $n$-th row and the $(n-1)$-th row of Pascal's Triangle?

In the diagram provided, a 'hexagon' is formed around the central value in Pascal's Triangle. According to the property of the triangle, the product of alternating neighbors of a cell is equal. If the neighbors are $a, b, c, d, e, f$, then $a \times c \times e = b \times d \times f$. What is the central value if its row is $n=4$ and it's the middle element?

In Pascal's Triangle, the entry in the $n^{th}$ row and $r^{th}$ column (starting from $0$) corresponds to the binomial coefficient $^nC_r$. Based on the property shown in the diagram where two adjacent nodes sum to the one below them, which identity is being illustrated?

If the sum of the first three coefficients in the expansion of $(1+x)^n$ is $22$, what is $n$?

What is the ratio of the coefficient of $x^3$ to $x^2$ in the expansion of $(1+x)^6$?

A computer science student is optimizing a search algorithm that uses values from Pascal's Triangle. They need to find the sum: $^{10}C_{0} + ^{11}C_{1} + ^{12}C_{2} + ... + ^{10+k}C_{k}$. If $k=5$, what is the value of this sum according to the Hockey-stick identity?

During a chemistry experiment, the concentration of a mixture is found to follow the coefficients of $(1+x)^n$. If the $4^{th}$, $5^{th}$, and $6^{th}$ coefficients are in arithmetic progression, find the smallest possible value of $n$.

A student notices that in the $7^{th}$ row of Pascal's Triangle, the sum of the elements is $128$. If they create a new sequence by multiplying each element $^{7}C_{r}$ by $2^r$, what is the sum of this new sequence?

How many terms are in the expansion of $(a + b)^{13}$?

The 4th term in the expansion of $(a - b)^4$ is:

What is the 3rd term in the expansion of $(x + 1)^8$?

The middle term of $(1 + x)^{22}$ is:

The coefficient of the middle term in the expansion of $(1+x)^{10}$ is:

Calculate the $8^{th}$ term of $(1-2x)^{12}$.

Find the term independent of $x$ in the expansion of $(x + \frac{1}{x^2})^6$.

In the expansion of $(3 + ax)^9$, the coefficients of $x^2$ and $x^3$ are equal. What is the value of $a$?

If the fourth term in the expansion of $(px + 1/x)^n$ is $5/2$, find $n$ and $p$ given $n$ is 6.

Find the coefficient of $x^4$ in the expansion of $(1+x+x^2+x^3+x^4)^2$.

What is the coefficient of $x^1$ in the expansion of $(1 + x)^n$?

In the expansion of $(x - 1)^6$, the coefficient of the middle term is:

Calculate $(10.1)^2$ using the expansion of $(10 + 0.1)^2$.

In the expansion of $(a + b)^4$, the coefficients are 1, 4, 6, 4, 1. These are also known as:

Find the coefficient of $x^3$ in the expansion of $(x^2 + 2x + 1)^3$.

The value of $\binom{n}{0} + 2\binom{n}{1} + 3\binom{n}{2} + \dots + (n+1)\binom{n}{n}$ is:

Find the term independent of $x$ in the expansion of $(x^3 - \frac{1}{x^2})^{10}$.

What is the remainder when $3^{2024}$ is divided by $10$?

Find the value of $(\sqrt{2} + 1)^6 + (\sqrt{2} - 1)^6$.

What is the coefficient of $x^k$ in the expansion of $1 + (1+x) + (1+x)^2 + \dots + (1+x)^n$?

What is the value of $\binom{3}{0} + \binom{3}{1} + \binom{3}{2} + \binom{3}{3}$?

What is the middle term of $(1+x)^2$?

How many terms are in the expansion of $(1+x)^{n-1}$?

What is the value of $\binom{5}{1}$?

If the coefficients of $x^7$ and $x^8$ in $(2 + \frac{x}{3})^n$ are equal, then $n$ is:

In the expansion of $(1+x)^{2n+1}$, the sum of the coefficients of the two middle terms is:

The sum of the numerical coefficients in the expansion of $(1-x+x^2)^n$ is:

Find the term independent of $x$ in the expansion of $(x + \frac{1}{x^2})^{12}$.

Evaluate the sum $\sum_{r=1}^n (-1)^{r-1} r \binom{n}{r}$.

If $n$ is even, the greatest value of $\binom{n}{r}$ occurs at $r$ equal to: