Q1. Given below are two statements :
A number of distinct 8-letter words are possible using the letters of the word SYLLABUS. If a word in chosen at random, then
Statement I: The probability that the word contains the two S’s together is 1/4.
Statement II : The probability that the word begins and ends with L is 1/28.
In the light of the above statements, choose the correct answer from the options given below
A Both Statement I and Statement II are true
B Both Statement I and Statement II are false
C Statement I is true but Statement II is false
Statement I is false but Statement II is true
EXPLANATION
Total number of 8-letter words possible using letters S, Y, L, L, A, B, U, S is 8! / 2! * 2!
Statement I:
We can find total number of 8-letter words by considering 2 S’s as a single unit and arranging 7 units, i.e. (SS), Y, L, L, A, B, U
Total number of ways = 7! / 2!
Required probability = 7!/2! / 8!/2!*2! = 2/8= 1/4
Therefore, statement I is correct.
Statement II:
We can find total number of 8-letter words such that L’s are at ends by arranging remaining 6 letters in the middle.
Total number of ways = 6!/2!
Required probability = 6!/2! / 8!/2!*2! = 2 / 56 = 1 / 28
Therefore, statement II is correct.
The answer is option A.
Q2. What is the probability of getting a sum of 22 or more when four dice are thrown?
A 5/432
B 4/648
C 1/144
D 7/36
EXPLANATION
Sum is 24: The only possibility is (6,6,6,6) -> 1 way
Sum is 23: The only possibility is (6,6,6,5) -> 4! / 3! = 4 ways
Sum is 22: There are two possibilities, i.e. (6,6,6,4) and (6,6,5,5)
(6,6,6,4) -> 4! / 3! = 4 ways
(6,6,5,5) -> 4! / 2!*2! = 6 ways
Total number of ways satisfying the given condition = 1 + 4 + 4 + 6 = 15
Total number of possibilities = (6)(6)(6)(6)
Required probability = 15 / 1296 = 5 / 432
The answer is option A
Q3. Given below are two statements :
Statement I: The number of ways to pack six copies of the same book into four identical boxes where a box can contain as many as six books, is 9.
Statement II: The minimum number of students needed in a class to guarantee that there are at least six students whose birthday fall in the same month, is 61 .
In the light of the above statements, choose the correct answer from the options given below.
A Both Statement I and Statement II are true
B Both Statement I and Statement II are false
C Statement I is true but Statement II is false
D Statement I is false but Statement II is true
EXPLANATION
Statement 1
We need to pack 6 indistinguishable books into 4 indistinguishable boxes
The ways we can pack the books are
6 ,0,0,0 (only one way as 0,6,0,0 or 0,0,6,0 won’t be different as boxes are identical)
5, 1,0,0
4, 2,0,0
4, 1, 1,0
3, 3,0,0
3, 2, 1,0
3, 1, 1, 1
2, 2, 2,0
2, 2, 1, 1.
so total 9 ways
Hence statement 1 is correct.
Statement 2
To calculate the minimum number of students required, we need to consider the worst case scenario i.e. the birthdays of students are evenly spread across all the 12 months. We’ll have to assume that there are atleast 5 birthday is each month, so the total number of students required = 5* 12 = 60. To assure that there is 1 month in which we have 6 birthdays, we’ll have to add 1 to this number, hence the minimum number of students required
= 60 + 1 = 61
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