1. In how many different ways can the letters of the word 'LEADING' be arranged in such a way that the vowels always come together?

**Answer: **720

**Explanation:**
The word 'LEADING' has 7 different letters.

When the vowels EAI are always together, they can be supposed to form one letter.

Then, we have to arrange the letters LNDG (EAI).

Now, 5 (4 + 1 = 5) letters can be arranged in 5! = 120 ways.

The vowels (EAI) can be arranged among themselves in 3! = 6 ways.

Required number of ways = (120 x 6) = 720.

2. In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?

**Answer: **63

**Explanation:**
Here is no explanation for this answer

3. In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?

**Answer: **50400

**Explanation:**
In the word "CORPORATION",treat those vowels OOAIO as one letter.

so it will be "CRPRTN (OOAIO)".

which is having 7 (6 + 1) letters. out of which R occurs 2 times and the rest are different.

Number of ways arranging these letters = (7!/2!) = 2520.

Now 5 vowels in which O occurs 3 times and the rest are different, can be arranged as 5!/3! = 20 ways.

So required number of ways = (2520 x 20) = 50400.

4. How many 4-letter words with or without meaning, can be formed out of the letters of the word, 'LOGARITHMS', if repetition of letters is not allowed?

**Answer: **5040

**Explanation:**
LOGARITHMS' contains 10 different letters.

Required number of words = Number of arrangements of 10 letters, taking 4 at a time.

= 10P4

= (10 x 9 x 8 x 7)

= 5040.

5. 23 people are there, they are shaking hands together, how many hand shakes possible, if they are in pair of cyclic sequence.

**Answer: **22

**Explanation:**
Here is no explanation for this answer

6. Six friends go to pizza corner there are 2 types of pizzas. And six different flavors are there they have to select 2 flavors from 6 flavors. In how many ways we can select?

**Answer: **6C2

**Explanation:**
Here is no explanation for this answer

7. 36 people {a1, a2, ..., a36} meet and shake hands in a circular fashion. In other words, there are totally 36 handshakes involving the pairs, {a1, a2}, {a2, a3}, ..., {a35, a36}, {a36, a1}. Then size of the smallest set of people such that the rest have shaken hands with at least one person in the set is

**Answer: **18

**Explanation:**
Here is no explanation for this answer

8. From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done?

**Answer: **756

**Explanation:**
We may have (3 men and 2 women) or (4 men and 1 woman) or (5 men only).

Required number of ways = (7C3 * 6C2)+(7C4 * 6C1)+(7C5)

=>[(7*6*5)/(3*2*1) * (6*5)/(2*1)]+(7C3 * 6C1) + (7C2)

=> 525 + [(7*6*5)/(3*2*1)* 6]+[(7*6)/(2*1)]

= (525 + 210 + 21)

= 756

9. In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together?

**Answer: **120960

**Explanation:**
In the word 'MATHEMATICS', we treat the vowels AEAI as one letter.

Thus, we have MTHMTCS (AEAI).

Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different.

So,Number of ways of arranging these letters = (8!/2!*2!)=10080.

Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.

Number of ways of arranging these letters = (4!/2)=12.

=> Required number of words = (10080 * 12) = 120960.

10. Consider the word ROTOR. Whichever way you read it, from left to right or from right to left, you get the same word. Such a word is known as palindrome. Find the maximum possible number of 5-letter palindromes?

**Answer: **17576

**Explanation:**
The first letter from the right can be chosen in 26 ways because there are 26 alphabets.

Having chosen this, the second letter can be chosen in 26 ways.

=> The first two letters can be chosen in 26*26=676 ways

Having chosen the first two letters, the third letter can be chosen in 26 ways.

=> All the three letters can be chosen in 676*26=17576 ways.

=> maximum possible number of five letter palindromes is 17576 because the fourth letter is the same as the second letter and the fifth letter is the same as the first letter.