Question Detail

How many words can be formed from the letters of the word "SIGNATURE" so that vowels always come together.

17280
4320
720
80

Answer: Option A

Explanation:

word SIGNATURE contains total 9 letters.
There are four vowels in this word, I, A, U and E

Make it as, SGNTR(IAUE), consider all vowels as 1 letter for now
So total letter are 6.
6 letters can be arranged in 6! ways = 720 ways
Vowels can be arranged in themselves in 4! ways = 24 ways

Required number of ways = 720*24 = 17280

1. Evaluate combination
$\begin{aligned} ^{100}{C}_{100} \end{aligned}$

10000
1000
10
1

Answer: Option D

Explanation:

$\begin{aligned} ^{n}{C}_{n} = 1 \\ ^{100}{C}_{100} = 1 \end{aligned}$

2. Evaluate permutation equation
$\begin{aligned} ^{59}{P}_3 \end{aligned}$

195052
195053
195054
185054

Answer: Option C

Explanation:

$\begin{aligned} ^n{P}_r = \frac{n!}{(n-r)!} \\ ^{59}{P}_3 = \frac{59!}{(56)!} \\ = \frac{59 * 58 * 57 * 56!}{(56)!} \\ = 195054 \end{aligned}$

3. In how many way the letter of the word "RUMOUR" can be arranged

2520
480
360
180

Answer: Option D

Explanation:

In above word, there are 2 "R" and 2 "U"
So Required number will be

$\begin{aligned} = \frac{6!}{2!*2!} \\ = \frac{6*5*4*3*2*1}{4} \\ = 180 \end{aligned}$

4. In a Cricket cup total 153 matches were played and every two teams played exactly one match with each other. So what were the total number of teams participating in Cricket Cup ?

Answer: Option D

Explanation:

Lets suppose there were x number os team in cricket cup then from the given statement we can calculate :

$\begin{aligned} => ^n C _2 = 153 \\ \text{ it given n = 18 and n = -17} \\ \end{aligned}$
As answer cannot be negative to 18 is the answer

5. How many words can be formed by using all letters of TIHAR

Answer: Option B

Explanation:

First thing to understand in this question is that it is a permutation question.
Total number of words = 5
Required number =

$\begin{aligned} ^5{P}_5 = 5! \\ = 5*4*3*2*1 = 120 \end{aligned}$

Question Detail

How many words can be formed from the letters of the word "SIGNATURE" so that vowels always come together.

1. Evaluate combination
$\begin{aligned} ^{100}{C}_{100} \end{aligned}$

2. Evaluate permutation equation
$\begin{aligned} ^{59}{P}_3 \end{aligned}$

3. In how many way the letter of the word "RUMOUR" can be arranged

4. In a Cricket cup total 153 matches were played and every two teams played exactly one match with each other. So what were the total number of teams participating in Cricket Cup ?

5. How many words can be formed by using all letters of TIHAR

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Question Detail

How many words can be formed from the letters of the word "SIGNATURE" so that vowels always come together.

1. Evaluate combination 100C100\begin{aligned} ^{100}{C}_{100} \end{aligned}

2. Evaluate permutation equation 59P3\begin{aligned} ^{59}{P}_3 \end{aligned}

3. In how many way the letter of the word "RUMOUR" can be arranged

4. In a Cricket cup total 153 matches were played and every two teams played exactly one match with each other. So what were the total number of teams participating in Cricket Cup ?

5. How many words can be formed by using all letters of TIHAR

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1. Evaluate combination
$\begin{aligned} ^{100}{C}_{100} \end{aligned}$

2. Evaluate permutation equation
$\begin{aligned} ^{59}{P}_3 \end{aligned}$