There are 8 boys and 7 girls in a group. For each of the tasks specified below, write an expression for the number of ways of doing it. Do NOT try to simplify your answers. a) Sitting in a row so that all boys sit contiguously and all girls sit contiguously, i.e., no girl sits between any two boys and no boy sits between any two girls
Answer: b) Sitting in a row so that between any two boys there is a girl and between any two girls there is a boy
Answer: c) Choosing a team of six people from the group
Answer: d) Choosing a team of six people consisting of unequal number of boys and girls
Answer:

A training center operates 5 different types of experience programs. Two participants A and B, who participated in the programs at this training center, each want to select 2 types from the 5 types of experience programs. Find the number of cases where A and B select exactly one type of experience program in common. [4 points]

A training center operates five different types of experience programs. Participants A and B each want to select 2 types from the 5 types of experience programs. Find the number of cases where A and B select exactly one type in common. [4 points]

From 2 female students and 4 male students, select 3 people to participate in a science and technology competition, with at least 1 female student selected. The total number of different selection methods is $\_\_\_\_$ (Answer with numerals)

Five volunteers participate in community service over Saturday and Sunday. Each day, two people are randomly selected from them to participate. The number of ways to select such that exactly one person participates on both days is
A. $120$
B. $60$
C. $40$
D. $30$

Let $A$ be the set $\{ 1,2 , \ldots , 20 \}$. Fix two disjoint subsets $S _ { 1 }$ and $S _ { 2 }$ of $A$, each with exactly three elements. How many 3-element subsets of $A$ are there, which have exactly one element common with $S _ { 1 }$ and at least one element common with $S _ { 2 }$?
(a) 51
(b) 102
(c) 135
(d) 153

A box contains 10 red cards numbered $1, \ldots, 10$ and 10 black cards numbered $1, \ldots, 10$. In how many ways can we choose 10 out of the 20 cards so that there are exactly 3 matches, where a match means a red card and a black card with the same number?
(A) $\binom { 10 } { 3 } \binom { 7 } { 4 } 2 ^ { 4 }$
(B) $\binom { 10 } { 3 } \binom { 7 } { 4 }$
(C) $\binom { 10 } { 3 } 2 ^ { 7 }$
(D) $\binom { 10 } { 3 } \binom { 14 } { 4 }$

A box contains 10 red cards numbered $1, \ldots, 10$ and 10 black cards numbered $1, \ldots, 10$. In how many ways can we choose 10 out of the 20 cards so that there are exactly 3 matches, where a match means a red card and a black card with the same number?
(A) $\binom{10}{3} \binom{7}{4} 2^4$
(B) $\binom{10}{3} \binom{7}{4}$
(C) $\binom{10}{3} 2^7$
(D) $\binom{10}{3} \binom{14}{4}$

A box contains 10 red cards numbered $1, \ldots, 10$ and 10 black cards numbered $1, \ldots, 10$. In how many ways can we choose 10 out of the 20 cards so that there are exactly 3 matches, where a match means a red card and a black card with the same number?
(A) $\binom { 10 } { 3 } \binom { 7 } { 4 } 2 ^ { 4 }$
(B) $\binom { 10 } { 3 } \binom { 7 } { 4 }$
(C) $\binom { 10 } { 3 } 2 ^ { 7 }$
(D) $\binom { 10 } { 3 } \binom { 14 } { 4 }$

A box contains 10 red cards numbered $1 , \ldots , 10$ and 10 black cards numbered $1 , \ldots , 10$. In how many ways can we choose 10 out of the 20 cards so that there are exactly 3 matches, where a match means a red card and a black card with the same number?
(a) $\binom { 10 } { 3 } \binom { 7 } { 4 } 2 ^ { 4 }$.
(B) $\binom { 10 } { 3 } \binom { 7 } { 4 }$.
(C) $\binom { 10 } { 3 } 2 ^ { 7 }$.
(D) $\binom { 10 } { 3 } \binom { 14 } { 4 }$.

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is
(A) 380
(B) 320
(C) 260
(D) 95

Let $S = \{ 1,2,3 , \ldots , 9 \}$. For $k = 1,2 , \ldots , 5$, let $N _ { k }$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N _ { 1 } + N _ { 2 } + N _ { 3 } + N _ { 4 } + N _ { 5 } =$
[A] 210
[B] 252
[C] 125
[D] 126

In a high school, a committee has to be formed from a group of 6 boys $M _ { 1 } , M _ { 2 } , M _ { 3 } , M _ { 4 } , M _ { 5 } , M _ { 6 }$ and 5 girls $G _ { 1 } , G _ { 2 } , G _ { 3 } , G _ { 4 } , G _ { 5 }$.
(i) Let $\alpha _ { 1 }$ be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls.
(ii) Let $\alpha _ { 2 }$ be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.
(iii) Let $\alpha _ { 3 }$ be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls.
(iv) Let $\alpha _ { 4 }$ be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls and such that both $M _ { 1 }$ and $G _ { 1 }$ are NOT in the committee together.
LIST-I P. The value of $\alpha _ { 1 }$ is Q. The value of $\alpha _ { 2 }$ is R. The value of $\alpha _ { 3 }$ is S. The value of $\alpha _ { 4 }$ is
LIST-II

1. 136
2. 189
3. 192
4. 200
5. 381
6. 461

The correct option is:
(A) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 6 ; ~ } \mathbf { R } \rightarrow \mathbf { 2 ; } \mathbf { S } \rightarrow \mathbf { 1 }$
(B) $\mathbf { P } \rightarrow \mathbf { 1 } ; \mathbf { Q } \rightarrow \mathbf { 4 } ; \mathbf { R } \rightarrow \mathbf { 2 } ; \mathbf { S } \rightarrow \mathbf { 3 }$
(C) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 6 } ; \mathbf { R } \rightarrow \mathbf { 5 } ; \mathbf { S } \rightarrow \mathbf { 2 }$
(D) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 2 } ; \mathbf { R } \rightarrow \mathbf { 3 } ; \mathbf { S } \rightarrow \mathbf { 1 }$

Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen ?
(A) 21816
(B) 85536
(C) 12096
(D) 156816

A committee of 4 persons is to be formed from 2 ladies, 2 old men and 4 young men such that it includes at least 1 lady, at least 1 old man and at most 2 young men. Then the total number of ways in which this committee can be formed is :
(1) 40
(2) 41
(3) 16
(4) 32

Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between them-selves exceeds the number of games that the men played with the women by 66 , then the number of men who participated in the tournament lies in the interval
(1) $( 11,13 ]$
(2) $( 14,17 )$
(3) $[ 10,12 )$
(4) $[ 8,9 ]$

A man $X$ has 7 friends, 4 of them are ladies and 3 are men. His wife $Y$ also has 7 friends, 3 of them are ladies and 4 are men. Assume $X$ and $Y$ have no common friends. Then the total number of ways in which $X$ and $Y$ together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of $X$ and $Y$ are in this party is:
(1) 485
(2) 468
(3) 469
(4) 484

A group of students comprises of 5 boys and n girls. If the number of ways, in which a team of 3 students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is 1750, then n is equal to
(1) 24
(2) 27
(3) 25
(4) 28

A committee of 11 member is to be formed from 8 males and 5 females. If $m$ is the number of ways the committee is formed with at least 6 males and $n$ is the number of ways the committee is formed with at least 3 females, then:
(1) $m = n = 68$
(2) $n = m - 8$
(3) $m = n = 78$
(4) $m + n = 68$

A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is $\_\_\_\_$

The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word 'SYLLABUS' such that two letters are distinct and two letters are alike, is

There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is:
(1) 3000
(2) 1500
(3) 2255
(4) 2250

A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is:
(1) 1050
(2) 1625
(3) 575
(4) 560

Team '$A$' consists of 7 boys and $n$ girls and Team '$B$' has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then $n$ is equal to:
(1) 5
(2) 2
(3) 4
(4) 6

There are ten boys $B _ { 1 } , B _ { 2 } , \ldots , B _ { 10 }$ and five girls $G _ { 1 } , G _ { 2 } , \ldots G _ { 5 }$ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both $B _ { 1 }$ and $B _ { 2 }$ together should not be the members of a group, is $\_\_\_\_$.