1
Numerical

### JEE Main 2021 (Online) 1st September Evening Shift

All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is ___________.

## Explanation

First find all possible words and then subtract words from each case that have both R together.

FARMER (6)

A, E, F, M, R, R

A
E
F A E
F A M
F A R E
F A R M E R

2
Numerical

### JEE Main 2021 (Online) 31st August Morning Shift

The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is ___________.

## Explanation

Total possible words = 6! = 720

When 4 consonants are together (V, W, L, S)
such cases = 3! ⋅ 4! = 144

All consonants should not be together

= Total $-$ All consonants together,

= 6! $-$ 3! 4! = 576
3
Numerical

### JEE Main 2021 (Online) 27th August Morning Shift

A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55, is ____________.

## Explanation

 5 a b b a 5

For divisible by 55 it shall be divisible by 11 and 5 both, for divisibility by 5 unit digit shall be 0 or 5 but as the number is six digit palindrome unit digit is 5.

A number is divisible by 11 if the difference between sum of the digits in the odd places and the sum of the digits in the even places is a multiple of 11 or zero.

Sum of the digits in the even place = a + b + 5

Sum of the digits in the odd places = a + b + 5

Difference between the two sums = (a + b + 5 ) - (a + b + 5) = 0

0 is divisible by 11.

Hence, 5abba5 is divisible by 11.

So, required number = 10 $\times$ 10 = 100
4
Numerical

### JEE Main 2021 (Online) 26th August Morning Shift

The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is ______________.

## Explanation

(i) When '0' is at unit place

Number of numbers = 20

(ii) When 4 or 6 are at unit place

Number of numbers = 32

Total three digit even number = 20 + 32 = 52