Question 1: From a group of cricket players, a team of
players is to be chosen. In how many ways can this be done?
Answer:
A group of cricket players, a team of
players is to be chosen in ways
Question 2: How many different boat parties of , consisting of
boys and
girls, can be made from
boys and
girls?
Answer:
Number of different boat parties of , consisting of
boys and
girls, that can be made from
boys and
girls
Question 3: In how many ways can a student choose courses out of
courses if
courses are compulsory for every student?
Answer:
courses are compulsory for every student.
Therefore courses are to be chosen from
possible courses.
Hence the number of ways a student can choose courses out of
courses if
courses are compulsory for every student
Question 4: In how many ways can a football team of players be selected from
players? How many of these will (i) include
particular players? (ii) exclude
particular players?
Answer:
Number of ways to select players from
players
i) If two particular players are always included, then the number of ways
ii) If two particular players are always excluded, then the number of ways
Question 5: There are professors and
students out of whom a committee of
professors and
students is to be formed. Find the number of ways in which this can be done. Further, find in how many of these committees:
(i) a particular professor is included.
(iii) a particular student is excluded.
(ii) a particular student is included.
Answer:
Number of ways committee can be formed
i) Number of ways committee can be formed if a particular professor is included
ii) Number of ways committee can be formed if a particular student is excluded
iii) Number of ways committee can be formed if a particular student is included
Question 6: How many different products can be obtained by multiplying two or more of the numbers 3, 5, 7, 11 (without repetition)?
Answer:
The number of ways products can be obtained by multiplying two or more of the number
Question 7: From a class of boys and
girls,
students are to be chosen for a competition; at least including
boys and
girls. The
girls who won the prizes last year should be included. In how many ways can the selection be made?
Answer:
Given girls who won the prizes last year should be included.
Therefore we have to select from boys and
girls. Therefore the combinations would be
Question 8: How many different selections of 4 books can be made from 10 different books, if (i) there is no restriction; (ii) two particular books are always selected (iii) two particular books are never selected?
Answer:
i) Number of ways in which books can be made from
different books if there is no restriction
ii) Number of ways in which books can be made from
different books if two particular books are always selected
iii) Number of ways in which books can be made from
different books if two particular books are never selected
Question 9: From officers and
soldiers in how many ways can
be chosen (i) to include exactly one officer (ii) to include at least one officer?
Answer:
Number of ways in which from officers and
soldiers in how many ways can 6 be chosen if there is no restriction
i) Number of ways in which from officers and
soldiers in how many ways can 6 be chosen to include exactly one officer
ii) Number of ways in which from officers and
soldiers in how many ways can 6 be chosen to include at least one officer
Question 10: A sports team of students is to be constituted, choosing at least
from class XI and at least
from class XII. If there are
students in each of these classes, in how many ways can the teams be constituted?
Answer:
The number of ways a sports team of students is to be constituted, choosing at least
from class XI and at least
from class XII from
students in each of these classes
Question 11: A student has to answer questions, choosing at least
from each of
and
. If there are
questions in
and
in
, in how many ways can the student choose
questions?
Answer:
Total number of questions
Questions in , Questions in
The number of ways the student choose questions
Question 12: In an examination, a student has to answer questions out of
questions; questions
and
are however compulsory. Determine the number of ways in which the student can make the choice.
Answer:
Total number of questions
Number of questions to be answered
Questions and
are however compulsory. Therefore the student has to choose
questions from the remainder
questions.
Number of ways to choose questions from the remainder
questions
Question 13: A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. In how many ways can he choose the 7 questions?
Answer:
Total number of questions
Number of questions to be answered
Number of ways can he choose the questions
Question 14: There are points in a plane of which
are collinear. How many different straight lines can be drawn by joining these points.
Answer:
Number of straight lines that can be drawn within points, taking
points at a time
Number of straight lines that can be drawn within collinear points, taking
points at a time
However, the collinear points will form a straight line.
Hence the total number of lines
Question 15: Find the number of diagonals of (i) a hexagon (ii) a polygon of sides.
Answer:
Note: A polygon has vertices. When you join two sides, you will either get a diagonal or a side.
i) Number of diagonals of a hexagon with sides
ii) Number of diagonals of a polygon with sides
Question 16: How many triangles can be obtained by joining 12 points, five of which are collinear?
Answer:
The number of triangles that can be obtained by joining points, five of which are collinear
Question 17: In how many ways can a committee of persons be formed out of
men and
women when at least one woman has to be necessarily selected ?
Answer:
The number of ways a a committee of persons be formed out of
men and
women when at least one woman has to be necessarily selected
Question 18: In a village, there are families of which
families have at most
children. In a rural development program,
families are to be helped chosen for assistance, of which at least
families must have at most
children. In how many ways can the choice be made ?
Answer:
There are families that have at most
children.
There are families that have more than
children.
Selection of families of which at least
families must have at most
children
Question 19: A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (i) no girl? (ii) at least one boy and one girl? (iii) at least 3 girls?
Answer:
Number of girls
Number of Boys
i) Number of ways members be selected if the team has no girl
ii) Number of ways members be selected if the team has at least
boy and
girls
iii) Number of ways members be selected if the team has at least
girls
Question 20: A committee of persons is to be constituted from a group of
men and
women. In how many ways can this be done? How many of these committees would consist of
man and
women?
Answer:
i) Number of ways committee can be selected
ii) Number of ways committee can be selected if consists of man and
women
Question 21: Find the number of (i) diagonals (ii) triangles formed in a decagon.
Answer:
Number of sides in a decagon
i) Number of diagonals
Question 22: Determine the number of cards combinations out of a deck of
cards if at least one of the
cards has to be a king ?
Answer:
Total number of cards in a deck
Number of kings in a deck
Therefore number of ways 5 cards combinations out of a deck of cards can be drawn if at least one of the
cards has to be a king
Question 23: We wish to select 6 persons from 8, but if the is chosen, then
must be chosen. In how many ways can the selection be made ?
Answer:
We need to select persons from
Number of ways persons can be selected when is selected
Number of ways persons can be selected when is not selected
Hence the total number of ways
Question 24: In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?
Answer:
Number of ways in which boys and
girls be selected from
boys and
girls
Question 25: Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each color.
Answer:
Number of ways of selecting balls from
red balls,
white balls and
blue balls if each selection consists of
balls of each color
Question 26: Determine the number of 5 cards combinations out of a deck of 52 cards if there is exactly one ace in each combination
Answer:
Total number of cards in a deck
Number of Aces in a deck
Therefore the number of ways in which cards combinations out of a deck of
cards if there is exactly one ace in each combination
Question 27: In how many ways can one select a cricket team of eleven from 17 players in which only 5 persons can bowl if each cricket team of 11 must include exactly 4 bowlers?
Answer:
Number of ways one can select a cricket team of eleven from players in which only
persons can bowl if each cricket team of
must include exactly
bowlers
Question 28: A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.
Answer:
The number of ways black and
red balls can be selected from a bag containing
black and
red balls
Question 29: In how many ways can a student choose a program of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student?
Answer:
The number of ways can a student can choose a program of courses if
courses are available and
specific courses are compulsory for every student
Question 30: A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consist of :
(i) exactly 3 girls? (ii) at least 3 girls? (iii) at most 3 girls?
Answer:
i) Number of ways to select the committee with exactly 3 girls
ii) Number of ways to select the committee with at least 3 girls
ii) Number of ways to select the committee with at most 3 girls
Question 31: In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?
Answer:
Number of ways the student can select the questions
Question 32: A parallelogram is cut by two sets of lines parallel to its sides. Find the number of parallelograms thus formed.
Answer:
In a parallelogram, there are two sets of parallel lines.
Now, each set of parallel lines are equal to lines.
Hence the number of parallelograms
Question 33: Out of 18 points in a plane, no three are in the same straight line except five points which are collinear. How many (i) straight lines (ii) triangles can be formed by joining them?
Answer:
Number of triangles formed