Permutation & Combination Solutions

54 curated permutation and combination problems covering core counting techniques, classic models, and complete worked solutions.

54 problems

Problem 1: Find the probability that is not in the first position

Four people stand in a line. Find the probability that is not in the first position.

Problem 2: Four volunteers each choose one of the projects

Four volunteers each choose one of the projects . Every project must be chosen by at least one volunteer, and volunteer is not allowed to choose project . How m

Problem 3: Two teachers and four students stand in a line for a photo

Two teachers and four students stand in a line for a photo. The two teachers must stand together, and that teacher-pair cannot occupy either end of the line. Ho

Problem 4: Arrange the six distinct letters in a row, subject to the conditions that is not in…

Arrange the six distinct letters in a row, subject to the conditions that is not in the first position and is not in the last position. How many arrangements ar

Problem 5: Four parents and three children stand in a line

Four parents and three children stand in a line. The first and last positions must both be occupied by parents. How many valid line-ups are there?

Problem 6: Two distinct students from each of classes stand in a row

Two distinct students from each of classes stand in a row. The two students from class must be adjacent, while the two students from class must not be adjacent.

Problem 7: Three boys, three girls, and a teacher stand in a row

Three boys, three girls, and a teacher stand in a row. The teacher must stand in the middle position, and among the three girls exactly one pair is adjacent. Ho

Problem 8: Six exhibition boards are arranged in a row

Six exhibition boards are arranged in a row. The boards labeled Rain Water and Grain Rain must not be adjacent, and the boards labeled White Dew and Cold Dew mu

Problem 9: Four cars are parked in five consecutive spaces

Four cars are parked in five consecutive spaces. Car occupies two consecutive spaces, and car must be parked adjacent to car . How many parking arrangements are

Problem 10: Two teachers and five students, including students and , stand in a row

Two teachers and five students, including students and , stand in a row. The two teachers must stand together, student must not stand next to a teacher, and stu

Problem 11: Four students are assigned to the three locations , with each location receiving at least one…

Four students are assigned to the three locations , with each location receiving at least one student. How many assignments are possible?

Problem 12: Seven rescue teams are assigned to five disaster counties, with each county receiving at least one…

Seven rescue teams are assigned to five disaster counties, with each county receiving at least one team. How many assignments are possible?

Problem 13: Six teachers are assigned to four different schools, with each school receiving at least one teacher

Six teachers are assigned to four different schools, with each school receiving at least one teacher. How many assignments are possible?

Problem 14: Ten people, consisting of six boys and four girls, are split into two unlabeled groups of…

Ten people, consisting of six boys and four girls, are split into two unlabeled groups of five each. The four girls are not allowed to all be in the same group.

Problem 15: Four duty groups are assigned to regions , with each region receiving at least one group

Four duty groups are assigned to regions , with each region receiving at least one group. One designated group is not allowed to go to region . How many assignm

Problem 16: The Zhao Shuang diagram is divided into five regions: one central square and four surrounding triangular…

The Zhao Shuang diagram is divided into five regions: one central square and four surrounding triangular regions. Four colors are available, and any two regions

Problem 17: A square pyramid is fixed in space

A square pyramid is fixed in space. Its five faces are colored using five available colors, and any two adjacent faces must have different colors. How many vali

Problem 18: Choose three digits from the odd digits and two digits from the even digits

Choose three digits from the odd digits and two digits from the even digits . Using the five chosen distinct digits, form five-digit numbers. How many such numb

Problem 19: Use the digits to form three-digit numbers with no repeated digit

Use the digits to form three-digit numbers with no repeated digit. A number is called peaked if its tens digit is greater than both its hundreds digit and its u

Problem 20: Find: (1) the total number of such numbers;

Using the digits without repetition, form four-digit numbers. Find: (1) the total number of such numbers; (2) how many are even; (3) how many are greater than .

Problem 21: Six cards are labeled

Six cards are labeled . Choose four of them and arrange the chosen cards into a four-digit number. How many different four-digit numbers can be formed?

Problem 22: Using the digits , form four-digit numbers with no repeated digit and with at most one…

Using the digits , form four-digit numbers with no repeated digit and with at most one even digit. How many such numbers are there?

Problem 23: A shelf initially has four items on the upper level and eight items on the lower…

A shelf initially has four items on the upper level and eight items on the lower level. Choose three items from the lower level and move them to the upper level

Problem 24: Seven ingredients are added to a pot one by one

Seven ingredients are added to a pot one by one. Mushrooms, bamboo shoots, and dried tofu must be added together as one batch; eggplant must be added after chic

Problem 25: Six students take a photo

Six students take a photo. The first four students have already lined up in a fixed relative order, and two more students join without changing that relative or

Problem 26: There are two stacks of containers, one with containers and one with containers

There are two stacks of containers, one with containers and one with containers. Each move removes the top container from one stack. In how many different order

Problem 27: Six dance performances are already arranged in a fixed order

Six dance performances are already arranged in a fixed order. Five singing performances are then inserted, and the relative order of the six dance performances

Problem 28: Sixteen identical places are distributed among three high-school grades, with each grade receiving at least one…

Sixteen identical places are distributed among three high-school grades, with each grade receiving at least one place. How many distributions are possible?

Problem 29: Ten roster spots are allocated among five classes for a basketball team, with each class contributing…

Ten roster spots are allocated among five classes for a basketball team, with each class contributing at least one student. Only the numbers from each class mat

Problem 30: Eleven preliminary-round places are distributed among four classes, and a class is allowed to receive zero…

Eleven preliminary-round places are distributed among four classes, and a class is allowed to receive zero places. How many distributions are possible?

Problem 31: Fifteen identical balls are placed into boxes

Fifteen identical balls are placed into boxes . Box must contain at least ball, box at least balls, and box at least balls. How many placements are possible?

Problem 32: Find both the number of positive integer solutions and the number of nonnegative integer solutions of

Find both the number of positive integer solutions and the number of nonnegative integer solutions of .

Problem 33: A dragon-boat squad has eight athletes: three can paddle only on the left, three can paddle…

A dragon-boat squad has eight athletes: three can paddle only on the left, three can paddle only on the right, and two can paddle on either side. Choose three a

Problem 34: Among eleven experts, five know only traditional Chinese medicine, four know only Western medicine, and two…

Among eleven experts, five know only traditional Chinese medicine, four know only Western medicine, and two know both. Select four experts for a traditional Chi

Problem 35: Five people can only sing, two people can only dance, and one person can do both

Five people can only sing, two people can only dance, and one person can do both. A performance needs two singers and two dancers. How many valid selections are

Problem 36: Among nine tour guides, three speak only English, four speak only Japanese, and two speak both…

Among nine tour guides, three speak only English, four speak only Japanese, and two speak both languages. Select six guides so that three serve as English guide

Problem 37: Among eleven athletes, five are good only at soccer, four are good only at basketball, and…

Among eleven athletes, five are good only at soccer, four are good only at basketball, and two are good at both. (1) Arrange all basketball-capable athletes in

Problem 38: From four distinct pairs of socks, three socks are chosen at random

From four distinct pairs of socks, three socks are chosen at random. What is the probability that exactly two of the chosen socks come from the same pair?

Problem 39: Three pairs of twins are present, so there are six people in total

Three pairs of twins are present, so there are six people in total. Two people are chosen, with the condition that they cannot come from the same twin pair. How

Problem 40: Two families, with seven people in total, line up for a test

Two families, with seven people in total, line up for a test. The two fathers must stand at the two ends of the line, and the three children must stand together

Problem 41: Find: (1) the number of selections that consist of exactly two full pairs;

From six distinct pairs of shoes, four individual shoes are chosen. Find: (1) the number of selections that consist of exactly two full pairs; (2) the number of

Problem 42: Two families, with eight people in total, rent two different cars, and each car seats four…

Two families, with eight people in total, rent two different cars, and each car seats four people. Each married couple must ride in the same car, and each coupl

Problem 43: A staircase has steps

A staircase has steps. Each move climbs either step or steps, and exactly moves are used. How many different climb patterns are possible?

Problem 44: A staircase has steps

A staircase has steps. Each move climbs either steps or steps. How many different climb patterns are possible?

Problem 45: Six identical vaulting boxes and three identical medicine-ball crates are distributed among teams

Six identical vaulting boxes and three identical medicine-ball crates are distributed among teams . Each team must receive at least one vaulting box, and the me

Problem 46: A cinema row has seven seats separated by an aisle: four seats on the left and…

A cinema row has seven seats separated by an aisle: four seats on the left and three on the right. Seven students take these seats. Xiaoming and Xiaogang must s

Problem 47: Among eight workers, four are chosen for a rest shift

Among eight workers, four are chosen for a rest shift. Workers Zhang and Li are not allowed to both be on rest, and Zhang and Wang must either both rest or both

Problem 48: Six differently colored balls are available

Six differently colored balls are available. Person simply chooses any three balls. Person first splits the six balls into two labeled piles and , each of size

Problem 49: Teachers are assigned to teach grades 1, 2, and 3, with each grade receiving at least…

Teachers are assigned to teach grades 1, 2, and 3, with each grade receiving at least one teacher. Teachers and are not allowed to teach grade 3, while teachers

Problem 50: Choose four different numbers from

Choose four different numbers from . How many 4-element subsets contain at least one pair of consecutive integers?

Problem 51: Principals from six schools are assigned to the four training locations Jiangsu, Hebei, Hubei, and Chongqing,…

Principals from six schools are assigned to the four training locations Jiangsu, Hebei, Hubei, and Chongqing, with each location receiving at least one principa