1

### JEE Main 2017 (Online) 9th April Morning Slot

The number of ways in which 5 boys and 3 girls can be seated on a round table if a particular boy B1 and a particular girl G1 never sit adjacent to each other, is :
A
5 $\times$ 6!
B
6 $\times$ 6!
C
7!
D
5 $\times$ 7!

## Explanation

Number of ways = Total - when B1 and G1 sit together

Total ways to seat 8 people on round table = (8 - 1)! = 7!

When B1 and G1 sit together then assume B1 and G1 are one people, so total 7 people are there and among B1 and G1 they can sit 2! ways.

So total no of ways when B1 and G1 sit together
= (7 - 1)! $\times$ 2! = 6! $\times$ 2!

Number of ways = 7! - 6! $\times$ 2! = 6!$\times$(7 - 2) = 5 $\times$ 6!
2

### JEE Main 2018 (Offline)

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is :
A
at least 750 but less than 1000
B
at least 1000
C
less than 500
D
at least 500 but less than 750

## Explanation

From 6 different novels 4 novels can be chosen = ${}^6{C_4}$ ways

And from 3 different dictionaries 1 can be chosen = ${}^3{C_1}$ ways

$\therefore$ From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary can be chosen = ${}^6{C_4} \times {}^3{C_1}$ ways

Let 4 novels are N1, N2, N3, N4 and 1 dictionary is D1.

Dictionary should be in the middle. So the arrangement will be like this

_ _ D1 _ _

On those 4 blank places 4 novels N1, N2, N3, N4 can be placed. And 4 novels can be arrange $4!$ ways.

$\therefore$ Total no of ways = ${}^6{C_4} \times {}^3{C_1}$$\times 4!$ = 1080
3

### JEE Main 2018 (Online) 15th April Morning Slot

n$-$digit numbers are formed using only three digits 2, 5 and 7. The smallest value of n for which 900 such distinct numbers can be formed, is :
A
6
B
7
C
8
D
9

## Explanation

In n digit number first place can be filled with any one of 2, 5, 7. So no of ways first digit can be filled = 3

Similarly,
no of ways 2nd digit can be filled = 3 ways
.
.
.
.
- - - - - - nth - - - - - - - = 3 ways

$\therefore$ Total numbers = 3 $\times$ 3 $\times$ 3 .... n times
= 3n
$\therefore$ According to question, for smallest value of n,

3n > 900

36 = 729 < 900

37 = 2187 > 900

$\therefore$ n = 7
4

### JEE Main 2018 (Online) 15th April Evening Slot

The number of four letter words that can be formed using the letters of the word BARRACK is :
A
120
B
144
C
264
D
270

## Explanation

Case 1 :

When all the four letters different then no of words
= 5C4 $\times$4!

Case 2 :

When out of four letters two letters are R and other two different letters are chosen from B, A, C, K then the no of words
= 4C2 $\times$ ${{4!} \over {2!}}$ = 72

Case 3 :

When out of four letters two letters are A and other two different letters are chosen from B, R, C, K then the no of words
= 4C2 $\times$ ${{4!} \over {2!}}$ = 72

Case 4 :

When word is formed using two R and two A then number of words
= ${{4!} \over {2!2!}}$ = 6

So, total number of 4 letters words possible = 120 + 72 + 72 + 6 = 270