Committee with gender/category constraints

Select a committee or group where the composition must satisfy constraints on the number of people from different categories (e.g., men/women, year groups, instrument types) such as 'at least 2 men', 'more women than men', or 'exactly 3 from each category'.

12 questions · Moderate -0.1

5.01a Permutations and combinations: evaluate probabilities
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CAIE S1 2021 June Q1
4 marks Moderate -0.3
1 A bag contains 12 marbles, each of a different size. 8 of the marbles are red and 4 of the marbles are blue. How many different selections of 5 marbles contain at least 4 marbles of the same colour?
CAIE S1 2023 June Q2
5 marks Standard +0.3
2
  1. Find the number of ways in which a committee of 6 people can be chosen from 6 men and 8 women if it must include 3 men and 3 women.
    A different committee of 6 people is to be chosen from 6 men and 8 women. Three of the 6 men are brothers.
  2. Find the number of ways in which this committee can be chosen if there are no restrictions on the numbers of men and women, but it must include no more than two of the brothers.
CAIE S1 2024 November Q5
6 marks Moderate -0.3
5 In a class of 21 students, there are 10 violinists, 6 guitarists and 5 pianists. A group of 7 is to be chosen from these 21 students. The group will consist of 4 violinists, 2 guitarists and 1 pianist.
  1. In how many ways can the group of 7 be chosen?
    On another occasion a group of 5 will be chosen from the 21 students. The group must contain at least 2 violinists, at least 1 guitarist and at most 1 pianist.
  2. In how many ways can the group of 5 be chosen?
CAIE S1 2003 June Q5
8 marks Moderate -0.8
5 A committee of 5 people is to be chosen from 6 men and 4 women. In how many ways can this be done
  1. if there must be 3 men and 2 women on the committee,
  2. if there must be more men than women on the committee,
  3. if there must be 3 men and 2 women, and one particular woman refuses to be on the committee with one particular man?
CAIE S1 2020 June Q4
4 marks Standard +0.8
4 In a music competition, there are 8 pianists, 4 guitarists and 6 violinists. 7 of these musicians will be selected to go through to the final. How many different selections of 7 finalists can be made if there must be at least 2 pianists, at least 1 guitarist and more violinists than guitarists?
CAIE S1 2014 June Q2
4 marks Standard +0.3
2 A school club has members from 3 different year-groups: Year 1, Year 2 and Year 3. There are 7 members from Year 1, 2 members from Year 2 and 2 members from Year 3. Five members of the club are selected. Find the number of possible selections that include at least one member from each year-group.
CAIE S1 2019 June Q4
6 marks Standard +0.3
4
  1. Find the number of ways a committee of 6 people can be chosen from 8 men and 4 women if there must be at least twice as many men as there are women on the committee.
  2. Find the number of ways a committee of 6 people can be chosen from 8 men and 4 women if 2 particular men refuse to be on the committee together.
CAIE S1 2014 November Q7
10 marks Standard +0.3
7 A committee of 6 people is to be chosen from 5 men and 8 women. In how many ways can this be done
  1. if there are more women than men on the committee,
  2. if the committee consists of 3 men and 3 women but two particular men refuse to be on the committee together? One particular committee consists of 5 women and 1 man.
  3. In how many different ways can the committee members be arranged in a line if the man is not at either end?
OCR MEI S1 2009 June Q2
5 marks Moderate -0.8
2 There are 14 girls and 11 boys in a class. A quiz team of 5 students is to be chosen from the class.
  1. How many different teams are possible?
  2. If the team must include 3 girls and 2 boys, find how many different teams are possible.
Edexcel FP2 2020 June Q1
6 marks Moderate -0.3
  1. A small sports club has 12 adult members and 14 junior members.
The club needs to enter a team of 8 players for a particular competition.
Determine the number of ways in which the team can be selected if
  1. there are no restrictions on the team,
  2. the team must contain 4 adults and 4 juniors,
  3. more than half the team must be adults.
Pre-U Pre-U 9794/3 2016 Specimen Q2
11 marks Moderate -0.8
2
  1. A music club has 200 members. 75 members play the piano, 130 members like Elgar, and 30 members do not play the piano, nor do they like Elgar.
    1. Calculate the probability that a member chosen at random plays the piano but does not like Elgar.
    2. Calculate the probability that a member chosen at random plays the piano given that this member likes Elgar.
  2. The music club is organising a concert. The programme is to consist of 7 pieces of music which are to be selected from 9 classical pieces and 6 modern pieces. Find the number of different concert programmes than can be produced if
    1. there are no restrictions,
    2. the programme must consist of 5 classical pieces and 2 modern pieces,
    3. there are to be more modern pieces than classical pieces.
Pre-U Pre-U 9794/3 2018 June Q6
12 marks Moderate -0.3
6 A volleyball squad has 11 players. A volleyball team consists of 6 players.
  1. Find the total number of different teams that could be chosen from the squad. The squad has 5 women and 6 men.
  2. Find the total number of different teams that contain at least 3 women. The squad includes a man and a woman who are married to one another.
  3. It is given that the team chosen has exactly 3 women and all such teams are equally likely to be chosen. Calculate the probability that a team chosen includes the married couple.