Topics / Counting & Probability

Arrangements with Restrictions

Permutations with constraints, adjacency, circular arrangements, forbidden positions

Arrangements with RestrictionsAMC 10 Step by Step

Name: Date: September 13, 2026

Counting & Probability · 25 problems have this as their main idea

primary-topic problems (1.9% of all)

more as a secondary topic

Where it appears

P1-10

P11-15

P16-20

P21-25

## What you need to know
- Block (glue) method: to keep $k$ specified objects together in a row of $n$ , treat them as one object: $(n-k+1)!\cdot k!$ arrangements.
- Gap method: to keep $k$ distinguishable objects pairwise non-adjacent, place the other $m$ objects first, creating $m+1$ gaps, then choose gaps: $\binom{m+1}{k}\,k!$ .
- Circular arrangements: $n$ distinct people around a table, rotations equivalent, sit in $(n-1)!$ ways; with reflections also equivalent, $\frac{(n-1)!}{2}$ . Labeled seats give $n!$ .
- Identical objects: $\frac{n!}{a!\,b!\cdots}$ .
- Derangements (nobody in their own spot): $D_2 = 1, D_3 = 2, D_4 = 9, D_5 = 44$ .

## How AMC 10 tests it
- Problems 8–15: seating in a row or around a table with "A and B must sit together" or "must not sit together."
- Letters of a word with a condition: vowels together, no two vowels adjacent, a given letter not first.
- Problems 15–22: restrictions interacting with symmetry (round table, necklaces, painted cubes) or "no two adjacent" on a circle.
- Often disguised as a probability: (restricted arrangements)/(all arrangements).

## Standard approaches
1. For "together," glue into a block; for "not together," subtract the together count from $n!$ .
2. For "no two of type X adjacent," place the others, count the gaps, choose gaps.
3. Seat the most constrained object first, then multiply the remaining choices; on a circle, fix one person to kill rotations.
4. Casework on the position of one special object, remembering that end seats have only one neighbor.
5. For $n \le 6$ , draw the slots and a small case tree.

## Worked example
Six students, including Ana and Ben, sit in a row of six chairs. Ana refuses to sit at either end, and Ben must sit next to Ana. How many seatings are possible?

(A) $96$ (B) $144$ (C) $192$ (D) $240$ (E) $288$

Solution. Ana takes one of the four interior chairs (positions $2$ through $5$ ). Each interior chair has two neighbors, so Ben has $2$ choices. The other four students fill the four remaining chairs in $4! = 24$ ways. Total $4 \cdot 2 \cdot 24 = 192$ , so the answer is $\boxed{\textbf{(C)}\ 192}$ .

## Pitfalls
- Forgetting to multiply by $k!$ for the internal order of a block.
- Using $(n-1)!$ for a circle when seats are labeled (then it is $n!$ ), or vice versa.
- Placing the restricted objects first in a "no two adjacent" problem instead of into the gaps.
- Handling "A not next to B" by position casework without noticing end seats have one neighbor, not two.

Traps that recur

Counting the labelled colourings and dividing by 8, which fails because three of the twelve classes are left fixed by a reflection; 84 is not even divisible by 8. (2025 AMC 10B #21)
Merging the two mirror-image spiral shapes into one because they look alike; a rotation can never turn a left-handed spiral into a right-handed one. (2021 AMC Fall 10B #24)
Trying to place numbers by trial without deriving a structural constraint, or dividing by the wrong symmetry count (24 rotations, not 48 symmetries). (2016 AMC 10A #18)
Counting the L/R patterns but then assigning the six shoes freely (3! * 3!) instead of noticing that each boundary forces a specific pair to occupy both boundary slots. (2024 AMC 10B #20)

Problems, easiest first

2018 AMC 10A · #4Arrangements with Restrictions

How many ways can a student schedule $3$ mathematics courses -- algebra, geometry, and number theory -- in a $6$ -period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other $3$ periods is of no concern here.)

2014 AMC 10A · #4Arrangements with Restrictions

Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?

2022 AMC 10A · #9Arrangements with Restrictions

A rectangle is partitioned into $5$ regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?

2015 AMC 10A · #10Arrangements with Restrictions

How many rearrangements of $abcd$ are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either $ab$ or $ba$ .

2012 AMC 10B · #11Arrangements with Restrictions

A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?

2025 AMC 10B · #16Arrangements with Restrictions

A circle has been divided into $6$ sectors of different sizes. Then $2$ of the sectors are painted red, $2$ painted green, and $2$ painted blue so that no two neighboring sectors are painted the same color. One such coloring is shown below. How many different colorings are possible?

2024 AMC 10B · #17Arrangements with Restrictions

In a race among $5$ snails, there is at most one tie, but that tie can involve any number of snails. For example, the result might be that Dazzler is first; Abby, Cyrus, and Elroy are tied for second; and Bruna is fifth. How many different results of the race are possible?

2022 AMC 10A · #14Arrangements with Restrictions

How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser number?

2021 AMC Fall 10A · #18Arrangements with Restrictions

A farmer's rectangular field is partitioned into $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want …

2017 AMC 10A · #19Arrangements with Restrictions

Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of 5 chairs under these conditions?

2000 AMC 10 · #13Arrangements with Restrictions

There are 5 yellow pegs, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 orange peg to be placed on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color?

2024 AMC 10B · #20Arrangements with Restrictions

Three different pairs of shoes are placed in a row so that no left shoe is next to a right shoe from a different pair. In how many ways can these six shoes be lined up?

2021 AMC 10A · #20Arrangements with Restrictions

In how many ways can the sequence $1, 2, 3, 4, 5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?

2021 AMC Fall 10A · #24Arrangements with Restrictions

Each of the $12$ edges of a cube is labeled $0$ or $1$ . Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the $6$ faces of the cube equal to $2$ ?

2018 AMC 10B · #18Arrangements with Restrictions

Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or …

2017 AMC 10B · #18Arrangements with Restrictions

In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?

2014 AMC 10B · #24Arrangements with Restrictions

The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is bad if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$ . Arrangements that differ only by a rotation or a reflection are considered the same. How …

2013 AMC 10B · #22Arrangements with Restrictions

The regular octagon $ABCDEFGH$ has its center at $J$ . Each of the vertices and the center are to be associated with one of the digits $1$ through $9$ , with each digit used once, in such a way that the sums of the numbers on the lines $AJE$ , $BJF$ , $CJG$ , and $DJH$ are all equal. In how many ways can this be done?

2012 AMC 10B · #22Arrangements with Restrictions

Let $(a_1,a_2, \dots ,a_{10})$ be a list of the first 10 positive integers such that for each $2 \le i \le 10$ either $a_i+1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list. How many such lists are there?

2011 AMC 10A · #22Arrangements with Restrictions

Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?

2010 AMC 10B · #23Arrangements with Restrictions

The entries in a $3 \times 3$ array include all the digits from $1$ through $9$ , arranged so that the entries in every row and column are in increasing order. How many such arrays are there?

2008 AMC 10B · #21Arrangements with Restrictions

Ten chairs are evenly spaced around a round table and numbered clockwise from $1$ through $10$ . Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arrangements are possible?

2025 AMC 10B · #21Arrangements with Restrictions

Each of the $9$ squares in a $3 \times 3$ grid is to be colored red, blue, or yellow in such a way that each red square shares an edge with at least one blue square, each blue square shares an edge with at least one yellow square, and each yellow square shares an edge with at least one red square. Colorings that can be …

2021 AMC Fall 10B · #24Arrangements with Restrictions

A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)

2016 AMC 10A · #18Arrangements with Restrictions

Each vertex of a cube is to be labeled with an integer $1$ through $8$ , with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. …