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AHL 1.10—Perms and combs, binomial with negative and fractional indices - IB Questionbank | RevisionDojo

Practice AHL 1.10—Perms and combs, binomial with negative and fractional indices with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

How many different ways can $4$ friends sit in a row of $4$ chairs?

[2]

Question 2

HLPaper 1

Expand $(3 + 2x)^{-3}$ in ascending powers of $x$ , up to and including the term in $x^2$ , simplifying the coefficients.

[4]

Question 3

HLPaper 1

When $(1 - 3x)(1 + ax)^{3/2}$ , where $a$ is a constant, is expanded in ascending powers of $x$ , the coefficient of the term in $x$ is zero.

Find the value of $a$ .

[3]

When $a$ has this value, find the term in $x^2$ in the expansion of $(1 - 3x)(1 + ax)^{3/2}$ , simplifying the coefficient.

[4]

Question 4

HLPaper 1

A university committee is to be formed from $15$ professors: $7$ from the Science faculty, and $8$ from the Arts faculty. The committee must have $6$ members.

In how many different ways can the committee be chosen if it must consist of $3$ professors from Science and $3$ professors from Arts?

[2]

The $6$ chosen committee members are to be arranged in a row for a photograph. In how many ways can they be arranged if the $3$ Arts professors chosen must stand next to each other?

[3]

In another arrangement, the $3$ chosen Science professors and $3$ chosen Arts professors are to stand in a single line. How many arrangements are possible if no two Science professors stand next to each other?

[4]

Question 5

HLPaper 1

A student needs to complete five different homework assignments: Math, Physics, Chemistry, English, and History.

In how many different orders can the student complete all five assignments?

[2]

If the student decides that Math must be completed first, in how many different orders can the remaining assignments be completed?

[2]

Question 6

HLPaper 1

Find the total number of different $3$ -digit numbers that can be formed using all the digits in the number $753$ .

[2]

Question 7

HLPaper 1

Expand $(1 - x)\sqrt[3]{(1 + 3x)}$ in ascending powers of $x$ , up to and including the term in $x^2$ , simplifying the coefficients.

[4]

Question 8

HLPaper 1

A committee is to be formed consisting of $1$ teacher and $2$ students. There are $8$ teachers and $10$ students available.

In how many different ways can the $2$ students be chosen?

[2]

How many different committees can be formed?

[2]

Question 9

HLPaper 1

Let $m,n\in\mathbb{N}$ and $k\in\{0,1,\dots,m+n\}$ .

Prove Vandermonde’s identity $\sum_{r=0}^k \binom{m}{r}\binom{n}{k-r}=\binom{m+n}{k}.$

[3]

Deduce $\displaystyle \sum_{r=0}^{n}\binom{n}{r}^2=\binom{2n}{n}$ .

[3]

Expand $(1-x)^{-3}$ up to and including $x^3$ .

[3]

Question 10

HLPaper 1

Let $n\in\mathbb{N}$ and $x$ be real.

Prove that $\sum_{r=0}^n \binom{n}{r}x^{r-1}=\frac{(1+x)^n-1}{x}\quad (x\neq0).$

[3]

Show that $\sum_{r=0}^n r\binom{n}{r}x^{r-1}=n(1+x)^{n-1}.$

[3]

Expand $(1-x)^{-4}$ up to and including $x^3$ .

[3]

AHL 1.10—Perms and combs, binomial with negative and fractional indices Questionbank