Permutation and Combinations MATHS - Permutation and Combinations

Six identical coins are arranged in a row.The number of ways in which the number of tails is equal to the number of heads is:

There are 5 roads leading to a town from a village.The number of different ways in which a villager can go to the town and return back, is:

The total number of ways in which 9 different boys can be distributed among three different children, so that the youngest gets 4, the middle gets 3 and the oldest gets 2, is:

In how many ways can 12 people be divided into 3 groups where 4 persons must be there in each group?

The number of different signals which can be given from 6 flags of different colours taking one or more at a time, is:

State True or False for the following statement:
In a steamer there are stalls for 12 animals, and there are horses, cows and calves (not less than 12 each) ready to be shipped. They can be loaded in 3¹² ways

State True or False for the following statement:
There are 12 points in a plane of which 5 points are collinear, then the number of lines obtained by joining these points in pairs is ¹²C₂ – ⁵C₂ .

State True or False for the following statement:
To fill 12 vacancies there are 25 candidates of which 5 are from scheduled castes. If 3 of the vacancies are reserved for scheduled caste candidates while the rest are open to all, the number of ways in which the selection can be made is ⁵C₃ × ²⁰C₉ .
In each if the Exercises from 60 to 64 match each item given under the column C1 to its correct answer given under the column C₂ .

State True or False for the following statement:
Three letters can be posted in five letterboxes in 35 ways.

Fill in the Blank.
The number of permutations of n different objects, taken r at a line, when repetitions are allowed, is ______.

Fill in the Blank.
If ⁿP_r = 840, ⁿC_r = 35, then r = ______.

Fill in the Blank.
A committee of 6 is to be chosen from 10 men and 7 women so as to contain atleast 3 men and 2 women. In how many different ways can this be done if two particular women refuse to serve on the same committee.
[Hint: At least 3 men and 2 women: The number of ways = ¹⁰C₃ × ⁷C₃ + ¹⁰C₄ × ⁷C₂ . For 2 particular women to be always there: the number of ways = ¹⁰C4 + ¹⁰C₃ × ⁵C₁ . The total number of committees when two particular women are never together = Total – together.]

Fill in the Blank.
A box contains 2 white balls, 3 black balls and 4 red balls. The number of ways three balls be drawn from the box if at least one black ball is to be included in the draw is ______.

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 consists of: atmost 3 girls?

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 consists of: atleast 3 girls?

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 consists of: exactly 3 girls?

How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if all letters are used but first letter is a vowel?

In how many ways can the letters of the word ASSASSINATION be arranged so that all the S's are together?

In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student?

A bag contains 5 black balls  and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected from the lot.

In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers?

Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.

It is required, to seat 5 men, and 4 women in a row so that the women occupy the even places. How many such arrangements are possible?

Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.

In an examination a question paper consist 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?

The English alphabet has 5 vowels and 21 consonants. How many words with  2 vowels and 2 different consonants can be formed from the alphabet?

How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which are divisible by 10 and, no digit is repeated?

Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: Number of rectangles on a chess board is ⁸C₂ × ⁸C₂.
Reason: To form a rectangle, we have to select any two of the horizontal line and any two of the vertical line.

1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
3. Assertion is correct statement but Reason is wrong statement.
4. Assertion is wrong statement but Reason is correct statement.

Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is ⁹C₃.
Reason: The number of ways of choosing any 3 places, from 9 different places is ⁹C₃.

1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
3. Assertion is correct statement but Reason is wrong statement.
4. Assertion is wrong statement but Reason is correct statement.

 Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:

Assertion: Product of five consecutive natural numbers is divisible by 4!.

Reason: Product of n consecutive natural numbers is divisible by (n + 1)!.

1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
3. Assertion is correct statement but Reason is wrong statement.
4. Assertion is wrong statement but Reason is correct statement. 

Five students Ajay, Shyam, Yojana, Rahul and Akansha are sitting in a playground in a line.

Based on the above information, answer the following questions.

(i) Total number of ways of sitting arrangement of five students is
    (a) 120     (b) 60     (c) 24     (d) None of these

(ii) Total number of arrangement of sitting, if Ajay and Yojana sit together, is
    (a) 60     (b) 48     (c) 72     (d) 120

(iii) Total number of arrangement 'Yojana and Rahul sitting at extreme position' is
    (a) 24     (b) 36     (c) 48     (d) 12

(iv) Total number of arrangement, if shyam is sitting in the middle, is
    (a) 24     (b) 12     (c) 6     (d) 36

(v) Total number of arrangement sitting Yojana and Rahul not sit together, is
    (a) 72     (b) 120     (c) 60     (d) 144