A group of 10 representatives is to be selected out of 12 seniors and 10 juniors. In how many different ways can the group be selected if it should have at least one senior ?

Correct Answer: C

Explanation:

It is explicitly given that all the 4 black balls are different, all the 3 red balls are different and all the 5 blue balls are different. Hence this is a case where all are distinct objects.

 

Initially let's find out the number of ways in which we can select the black balls. Note that at least 1 black ball must be included in each selection.

 

Hence, we can select 1 black ball from 4 black balls
or 2 black balls from 4 black balls.
or 3 black balls from 4 black balls.
or 4 black balls from 4 black balls.

 

Hence, number of ways in which we can select the black balls

 

= 4C1 + 4C2 + 4C3 + 4C4
= 2 4 - 1 ........(A)

 

Now let's find out the number of ways in which we can select the red balls. Note that at least 1 red ball must be included in each selection.

 

Hence, we can select 1 red ball from 3 red balls
or 2 red balls from 3 red balls
or 3 red balls from 3 red balls

 

Hence, number of ways in which we can select the red balls
= 3C1 + 3C2 + 3C3
= 2 3 - 1........(B)

 

Hence, we can select 0 blue ball from 5 blue balls (i.e, do not select any blue ball. In this case, only black and red balls will be there)
or 1 blue ball from 5 blue balls
or 2 blue balls from 5 blue balls
or 3 blue balls from 5 blue balls
or 4 blue balls from 5 blue balls
or 5 blue balls from 5 blue balls.

 

Hence, number of ways in which we can select the blue balls
= 5C0 + 5C1 + 5C2 + ? + 5C5
= 2 5..............(C)

 

From (A), (B) and (C), required number of ways
=   2 5 2 4 - 1 2 3 - 1

Correct Answer: 147320

Explanation:

Sum of 4 digit numbers = (2+4+6+8) x  P 3 3 x (1111) = 20 x 6 x 1111 = 133320 

Sum of 3 digit numbers = (2+4+6+8) x  P 2 3 x (111) = 20 x 6 x 111 = 13320 

Sum of 2 digit numbers = (2+4+6+8) x  P 1 3 x (11) = 20 x 3 x 11 = 660 

Sum of 1 digit numbers = (2+4+6+8) x  P 0 3 x (1) = 20 x 1 x 1 = 20 

 

Adding All , Sum = 147320

Correct Answer: 23

Explanation:

As in this problem , buying any fruit is different case , as buying apple is independent from buying banana. so ADDITION rule will be used.

 

C 1 9 + C 1 8 + C 1 6 = 23 will be answer.

Correct Answer: 32640

Explanation:

We know that zero can't be in hundreds place. But let's assume that our number could start with zero.

 

The formula to find sum of all numbers in a permutation is

 

111 x no of ways numbers can be formed for a number at given position x sum of all given digits

 

No of 1 s depends on number of digits

 

So,the answer us

 

111 x 20 x (0+1+2+3+4+5) = 33300

 

We got 20 as follows. If we have 0 in units place we can form a number in 4*5 ways. This is for all numbers. So we have substituted 20 in formula.

 

Now, this is not the final answer because we have included 0 in hundreds place. so we have to remove the sum of all numbers that starts with 0.

 

This is nothing but the sum of all 2 digits numbers formed by 1 2 3 4 5. Because 0 at first place makes it a 2 digit number.

 

So the sum for this is 11 x 4 x (1+2+3+4+5).
=660

 

Hope u understood why we use 4. Each number can be formed in 4x1 ways

 

So, the final answer is 33300-660 = 32640

Correct Answer: None of these

Explanation:

100 C 10 + 90 C 20 + 70 C 30 + 40 C 40 = 100 ! 10 ! x 20 ! x 30 ! x 40 !

Correct Answer: 9!/(2!)^{2}x3!

Explanation:

n items of which p are alike of one kind, q alike of the other, r alike of another kind and the remaining are distinct can be arranged in a row in n!/p!q!r! ways.
The letter pattern 'MESMERISE' consists of 10 letters of which there are 2M's, 3E's, 2S's and 1I and 1R.
Number of arrangements =  9 ! ( 2 ! ) 2 × 3 !

Correct Answer: 600

Explanation:

Two men, three women and one child can be selected in ?C? x ?C? x ?C? ways = 600 ways.

Correct Answer: 5040

Explanation:

Given word is GLACIOUS has 8 letters.

=> C is fixed in one of the 8 places

Then, the remaining 7 letters can be arranged in 7! ways = 5040.

Correct Answer: 60

Explanation:

Here given the required digit number is 4 digit.

It must be divisible by 5. Hence, the unit's digit in the required 4 digit number must be 0 or 5. But here only 5 is available.

x x x 5

The remaining places can be filled by remaining digits as 5 x 4 x 3 ways.

 

Hence, number 4-digit numbers can be formed are 5 x 4 x 3 = 20 x 3 = 60.

Correct Answer: 1/5525

Explanation:

n(S) = 52C3 = 132600/6 = 22100

n(E) = 4C3 = 24/6 = 4