Robert Bosch Permutation and Combinations Quiz 3

NUMBER is 6 letters.
We have 4 places where letters are to be placed.
For first letter there are 6 choices, since repetition is not allowed, for second, third and fourth letter also we have 5, 4, and 3 choices resp, so total of 6*5*4*3 ways = 360 ways.

ALLIGATION contains 10 letters, so total 10! ways. There are 2 As, 2 Ls, 2 Is
So \frac{10!}{(2!*2!*2!)}

Take vowels in a box together as one – IIU, M, N, M, M
So there are 5 that to be placed for this 5!, now 3 Ms, so 5!/3!, so arrangement of vowels inside box gives \frac{3!}{2!}
So total = \frac{5!}{3!}* \frac{3!}{2!}

Total ways = ^{8}C_{4}*^{5}C_{3} = 140

A number is divisible by 4 when its last two digits are divisible by 4
For this the numbers should have their last two digits as 00, 04, 08, 12, 16, … 96
By the formula, an = a + (n-1)d
96 = 0 + (n-1)*4
n = 25
so there are 25 choices for last 2 digits and 9 choices (1-9) for the 1st digit
so total 9*25

0 cannot be placed at first digit to make it a 3 digit number.
3 cases:
Case 1: 2 is placed at first place
1 choice for the first place, 9 choices each for the 2nd and 3rd digit (0-9 except 2)
So numbers = 1*9*9 = 81
Case 2: 2 is placed at second place
8 choices for the first place (1-9 except 2), 1 choice for the 2nd digit and 9 choices for the 3rd digit (0-9 except 2)
So numbers = 8*1*9 = 72
Case 3: 2 is placed at third place
8 choices for the first place (1-9 except 2), 9 choices for the 2nd digit (0-9 except 2) and 1 choice for the 3rd digit
So numbers = 8*9*1 = 72
So total numbers = 81+72+72 = 225

 

Total 15 people, and a particular woman is to be taken to form a group of 5, so choice is to be done from
14 people of 4 people
Ways are ^{14}C_{4} = 1001

There are total 13 people, a particular man is to be excluded, so now 12 people are left to chosen from and
4 members to be chosen. So ways are ^{12}C_{4} .

0 cannot be on first place for it to be a 4 digit number,
So for 1st digit 4choices, for second also 4 (because 0 can be placed here), then 3 for third place, 2 for fourth place
Total numbers = 4*4*3*2

For 1st prize there are 8 choices, for 2nd prize, 7 choices, and for 3rd prize – 6 choices left
So total ways = 8*7*6