MAT-style counting, sequences and number questions

Arrangements, paths on grids, sequences and recurrences, and simple number theory. Many of these reward a careful systematic list. These are original problems in the style of the MAT. The MAT is no longer used for Oxford maths (from 2026 Oxford uses the TMUA), so treat them as problem-solving practice for the TMUA, STEP and interviews.

Practice questions

Try each one before you open a hint. The multiple-choice questions should take a few minutes each; give the longer ones 20 minutes or more.

Question 1: Keeping the As apart · multiple choice

In how many arrangements of the seven letters of ALGEBRA are the two As not next to each other?

  1. \(1440\)
  2. \(1800\)
  3. \(2160\)
  4. \(2520\)
Hint 1

Count all arrangements, remembering the repeated A.

Hint 2

Glue the two As together to count the arrangements where they touch.

Full solution

Answer: B: \(1800\)

All: \(\tfrac{7!}{2!}=2520\). With the As together (one block, six items): \(6!=720\). So \(2520-720=1800\).

Question 2: An arithmetic sum · multiple choice

An arithmetic sequence has third term \(7\) and eighth term \(22\). What is the sum of its first \(20\) terms?

  1. \(560\)
  2. \(590\)
  3. \(610\)
  4. \(620\)
Hint 1

Five steps of the common difference take you from the 3rd to the 8th term.

Hint 2

\(S_n=\tfrac n2\left(2a+(n-1)d\right)\).

Full solution

Answer: B: \(590\)

\(5d=15\), so \(d=3\) and \(a=7-6=1\). \(S_{20}=10(2+57)=590\).

Question 3: Paths on a grid · longer question

A path goes from \((0,0)\) to \((4,4)\) in steps of one unit, each step either right (R) or up (U).

  1. How many paths are there?
  2. How many paths do not pass through \((2,2)\)?
  3. How many paths never go above the line \(y=x\) (they may touch it)?
Hint 1

A path is a word with four Rs and four Us.

Hint 2

For (iii), list by the first return to the diagonal, or use the reflection trick: a bad path first touches \(y=x+1\); reflect the part before that touch.

Full solution

(i) \(\binom84=70\).

(ii) Through \((2,2)\): \(\binom42\cdot\binom42=36\). So \(70-36=34\).

(iii) Reflect the part of a bad path up to its first touch of \(y=x+1\) in that line: this matches bad paths with all paths from \((-1,1)\) to \((4,4)\), of which there are \(\binom83=56\). So \(70-56=14\).

Results: \(70\); \(34\); \(14\).

2 more multiple-choice and 1 more longer question

A last digit; Exactly three divisors; A recurrence with a linear term. Each with two hints and a full solution.

Included with A Level plans, and with IB, IGCSE or CBSE plans.

Next steps

Other areas: Algebra and functions · Calculus and graphs · Geometry and logic. Then: TMUA practice · STEP topic guides · Strategies

The MAT was set by the University of Oxford. A Level Math Revision is independent: it is not affiliated with or endorsed by the University of Oxford or any university. Every problem here is our own, written in the style of the MAT; for real papers use Oxford's own past papers page.