Permutations: Permutations are the different arrangements of a given number of things by taking some or all at a time.

Examples All permutations (or arrangements) that can be formed with the letters a, b, c by taking three at a time are (abc, acb, bac, bca, cab, cba) All permutations (or arrangements) that can be formed with the letters a, b, c by taking two at a time are (ab, ac, ba, bc, ca, cb) The number of permutations of n objects taken r at a time is determined by the following formula: P(n,r)=n!/(n−r)!

Combinations: Each of the different groups or selections formed by taking some or all of a number of objects is called a combination.

Examples: Suppose we want to select two out of three girls P, Q, R. Then, possible combinations are PQ, QR, and RP. (Note that PQ and QP represent the same selection.) Suppose we want to select three out of three girls P, Q, R. Then, the only possible combination is PQR The number of Combinations of n objects taken r at a time is determined by the following formula: nCr = n! / [r! * (n-r)!]

Repetition: The term repetition is very important in permutations and combinations. Consider the same situation described above where we need to find out the total number of possible samples of two objects which can be taken from three objects P, Q, R. If repetition is allowed, the same object can be taken more than once to make a sample. i.e., PP, QQ, RR can also be considered as possible samples. If repetition is not allowed, then PP, QQ, RR cannot be considered as possible samples

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Question 1 of 15

1. Question

How many unique ways are there to arrange the letters in the word PRIOR?

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Question 2 of 15

2. Question

How many ways can you arrange your reindeer?

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Question 3 of 15

3. Question

In how many ways can three boys can be seated on five chairs?

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Question 4 of 15

4. Question

How many numbers between 1 and 100 (inclusive) are divisible by 5 or 3?

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Question 5 of 15

5. Question

In how many ways can 5 different toys be packed in 3 identical boxes such that no box is empty, if any of the boxes may hold all of the toys?

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Question 6 of 15

6. Question

What is the value of 1×1!+2×2!+3×3!+…………n×n!; where n! means n factorial or n(n−1)(n−2)…1

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Question 7 of 15

7. Question

When six fair coins are tossed simultaneously, in how many of the outcomes will at most three of the coins turn up as heads?

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Question 8 of 15

8. Question

How many number of times will the digit 7 be written when listing the integers from 1 to 1000?

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Question 9 of 15

9. Question

How many words of 4 consonants and 3 vowels can be made from 12 consonants and 4 vowels, if all the letters are different?

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4 consonants out of 12 can be selected in 12C4 ways. 3 vowels can be selected in 4C3 ways. Therefore, total number of groups each containing 4 consonants and 3 vowels = 12C4 * 4C3 Each group contains 7 letters, which can be arranging in 7! ways. Therefore required number of words = 124 * 4C3 * 7!

Question 10 of 15

10. Question

How many different words can be formed with the letters of the word ‘SUPER’ such that the vowels always come together?

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Question 11 of 15

11. Question

Find the number of permutations of the letters of the word ‘REMAINS’ such that the vowels always occur in odd places.

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Question 12 of 15

12. Question

The number of straight lines that can be drawn out of 12 points of which 8 are collinear is

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Question 13 of 15

13. Question

A box contains three white balls, four black balls and three red balls. The number of ways in which three balls can be drawn from the box so that at least one of the balls is black is

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Question 14 of 15

14. Question

25 buses are running between two places P and Q. In how many ways can a person go from P to Q and return by a different bus?

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Question 15 of 15

15. Question

In how many different ways can 5 girls and 5 boys form a circle such that the boys and the girls alternate?