This set of Aptitude Questions and Answers (MCQs) focuses on “Permutations and Combinations – Set 5”.
1. A bag consists of 2 red balls, 3 green balls and 4 blue balls. In how many ways can 3 balls be drawn, if at least one of them is a blue ball?
a) 74
b) 76
c) 2! * 3! * 4!
d) 78
View Answer
Explanation: There are 2 red balls, 3 green balls and 4 blue balls.
The possible outcomes for at least one blue ball, when 3 balls are drawn from the bag are:
(1 red, 1 green, 1 blue), (1 red, 2 blue), (1 green, 2 blue), (2 red, 1 blue), (2 green, 1 blue), (3 blue).
Required number of ways = (2C1 * 3C1 * 4C1) + (2C1 * 4C2) + (3C1 * 4C2) + (2C2 * 4C1) + (3C2 * 4C1) + (4C3).
⟹ (2 * 3 * 4) + (2 * 6) + (3 * 6) + (1 * 4) + (3 * 4) + (4)
⟹ 24 + 12 + 18 + 4 + 12 + 4 = 74
Number of ways if at least one of them is blue = 74.
2. Three coins are tossed at a time. What is the probability of getting two heads?
a) 3 / 8
b) 1 / 2
c) 7 / 8
d) 2 / 3
View Answer
Explanation: When three coins are tossed, total number of outcomes will be as follows:
(HHH, HHT, HTH, THH, TTT, TTH, THT, HTT) = 8.
We have to find the probability of getting at least one head.
Required type of outcomes = (HHH, HHT, HTH, THH).
Probability of getting at least one head = 4 / 8 = 1 / 2.
3. How many three digit numbers can be formed with the numbers 8, 2, 4, 5, 1 (if repetition of digits is not allowed)?
a) 3!
b) 5!
c) 60
d) 80
View Answer
Explanation: The numbers given are 8, 2, 4, 5, 1.
Here, repetition of digits is not allowed.
The first digit can be arranged in 5 ways.
Second digit can be arranged in 4 ways.
Third digit can be arranged in 3 ways.
Therefore, Number of three digit numbers can be formed = 5 * 4 * 3 = 60.
4. Find the number of new words that can be formed with the word ‘AMBITION’ all starting with A.
a) 8! / 2!
b) 8! * 2!
c) 7!
d) 7! / 2!
View Answer
Explanation: The word ‘AMBITION’ has 8 letters.
The word has 1’A’, 1’M’, 1’B’, 2I’s, 1’T’, 1’O’ and 1’N’.
We have 8 letters that can be arranged in 8! Ways.
The letter A is fixed as the starting letter. So, 7 letters should be arranged in 7! Ways.
The letter ‘I’ is repeated = 2!.
Number of words that can be formed with starting letter as ‘A’ = 7! / 2!.
5. How many numbers can be formed with numbers 1, 2, 4, 5, 8 without repetition?
a) 125
b) 5!
c) 325
d) 625
View Answer
Explanation: Number of one digit numbers we can form = 5.
Number of two digit numbers we can form = 5 * 4 = 20.
Number of three digit numbers we can form = 5 * 4 * 3 = 60.
Number of four digit numbers we can form = 5 * 4 * 3 * 2 = 120.
Number of five digit numbers we can form = 5 * 4 * 3 * 2 * 1 = 120.
Total number of numbers we can form = 5 + 20 + 60 + 120 + 120 = 325.
6. How many three digit numbers can be formed such that all the digits are even?
a) 5!
b) 125
c) 5! * 3!
d) 100
View Answer
Explanation: We have 5 even numbers, they are 0, 2, 4, 6, 8.
For the first digit we have 4 options 2, 4, 6, 8.
For the second digit we have 5 options 0, 2, 4, 6, 8.
For the third digit we have 5 options 0, 2, 4, 6, 8.
Hence, number of three digit numbers that can be formed with even numbers = 4 * 5 * 5 = 100.
7. From a pack of cards, half of the cards are selected. What will be the possibility of outcomes?
a) 52! / 26!
b) 1
c) 52C25
d) 52C26
View Answer
Explanation: Pack of cards has 52 cards in it.
Half of the cards are selected.
Half of 52 cards is 26.
Selecting 26 out of 52 = 52C26.
The possible outcomes will be = 52C26.
8. In how many ways can the word ‘ESPECIALLY’ be arranged such that the vowels always come together?
a) 42600
b) 25200
c) 10080
d) 30240
View Answer
Explanation: The word ‘ESPECIALLY’ has 10 letters.
We have to consider the vowels (EEIA) as 1 letter.
Now, we have SPCLLY and (EEIA) i.e., 7(6 + 1) letters.
In SPCLLY there are 1 S, 1 P, 1 C, 2L’s and 1 Y.
Number of ways we can arrange these letters = 7! / 2! = 2520.
There are 4 vowels, can be arranged = 4! / (2!)(1!)(1!) = 12.
Number of ways we can arrange so that vowels can come together = (2520 * 12) = 30240.
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