Directions: Each of the questions given below is followed by series of options among which you have to choose the best one. You can also review your choices by clicking into answer section.
16. Naresh has 10 friends and he wants to invite 6 of them to a party. How many times will 3 particular fiends never attend the party?
A. 8
B. 720
C. 35
D. 7
Answer
Explanation:
Remove the 3 particular friends. This can only be done in 1 way —(A.
Now he needs to invite 6 friends from the remaining 7 (=10-3) friends. This can be done in 7C6 ways — (B.
From (A. and (B., required number of ways = 1 × 7C6 = 7C6
= 7C1 [∵ nCr = nC(n – r)]
= 7
17. If nP4: nP5= 1 : 2, then n =
A. 4
B. 6
C. 5
D. 7
Answer
Explanation:
(n! / (n – 4)!) *((n – 5)!/n!) = 1/2
n – 4 = 2
n= 6
18. In how many ways can 10 books be arranged on a shelf such that a particular pair of books should always be together?
A. 9! × 2!
B. 9!
C. 10! × 2!
D. 10!
Answer
Explanation:
We have a total of 10 books. Given that a particular pair of books should always be together. Hence, just tie these two books together and consider as a single book.
Hence we can take total number of books as 9. These 9 books can be arranged in 9P9 = 9! Ways.
We had tied two books together. These books can be arranged among themselves in 2P2 = 2! Ways. Hence, the required number of ways = 9! × 2!
19. There are 5 Rock songs, 6 Carnatic songs and 3 Indi pop songs. How many different albums can be formed using the above repertoire if the albums should contain at least 1 Rock song and 1 Carnatic song?
A. 15624
B. 16384
C. 6144
D. 240
Answer
Explanation:
There are 2n ways of choosing ‘n’ objects. For e.g. if n = 3, then the three objects can be chosen in the following 23 ways – 3C0 ways of choosing none of the three, 3C1 ways of choosing one out of the three, 3C2 ways of choosing two out of the three and 3C3 ways of choosing all three.
In the given problem, there are 5 Rock songs. We can choose them in 25 ways. However, as the problem states that the case where you do not choose a Rock song does not exist (at least one rock song has to be selecteD., it can be done in 25 – 1 = 32 – 1 = 31 ways.
Similarly, the 6 Carnatic songs, choosing at least one, can be selected in 26 – 1 = 64 – 1 = 63 ways.
And the 3 Indi pop can be selected in 23 = 8 ways. Here the option of not selecting even one Indi Pop is allowed.
Therefore, the total number of combinations = 31 * 63 * 8 = 15624
20. How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?
A.5
B.10
C.15
D.20
Answer
Explanation:
Since each desired number is divisible by 5, so we must have 5 at the unit place. So, there is 1 way of doing it.
The tens place can now be filled by any of the remaining 5 digits (2, 3, 6, 7, 9). So, there are 5 ways of filling the tens place.
The hundreds place can now be filled by any of the remaining 4 digits. So, there are 4 ways of filling it.
Required number of numbers = (1 x 5 x 4) = 20.
21. How many alphabets need to be there in a language if one were to make 1 million distinct 3 digit initials using the alphabets of the language?
A. 26
B. 50
C. 100
D. 1000
Answer
Explanation:
1 million distinct 3 digit initials are needed.
Let the number of required alphabets in the language be ‘n’.
Therefore, using ‘n’ alphabets we can form n * n * n = n3 distinct 3 digit initials.
Note: Distinct initials are different from initials where the digits are different. For instance, AAA and BBB are acceptable combinations in the case of distinct initials while they are not permitted when the digits of the initials need to be different.
This n3 different initials = 1 million
i.e. n3 = 106 (1 million = 106)
n3 = (102)3 => n = 102 = 100. Hence, the language needs to have a minimum of 100 alphabets to achieve the objective.
