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 number 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