Question Detail

In how many ways can the letters of the word "CORPORATION" be arranged so that vowels always come together.

5760
50400
2880
None of above

Answer: Option B

Explanation:

Vowels in the word "CORPORATION" are O,O,A,I,O
Lets make it as CRPRTN(OOAIO)

This has 7 lettes, where R is twice so value = 7!/2!
= 2520

Vowel O is 3 times, so vowels can be arranged = 5!/3!

= 20

Total number of words = 2520 * 20 = 50400

1. In how many words can be formed by using all letters of the word BHOPAL

Answer: Option D

Explanation:

Required number
\begin{aligned}
= 6! \\
= 6*5*4*3*2*1 \\
= 720
\end{aligned}

2. 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}

3. Evaluate permutation
\begin{aligned}
^5{P}_5
\end{aligned}

Answer: Option A

Explanation:

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

\end{aligned}

4. 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}

5. Evaluate permutation equation
\begin{aligned} ^{75}{P}_2\end{aligned}

5200
5300
5450
5550

Answer: Option D

Explanation:

\begin{aligned}
^n{P}_r = \frac{n!}{(n-r)!} \\
^{75}{P}_2 = \frac{75!}{(75-2)!} \\
= \frac{75*74*73!}{(73)!} \\
= 5550

\end{aligned}

Question Detail

In how many ways can the letters of the word "CORPORATION" be arranged so that vowels always come together.

1. In how many words can be formed by using all letters of the word BHOPAL

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

3. Evaluate permutation \begin{aligned} ^5{P}_5 \end{aligned}

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

5. Evaluate permutation equation \begin{aligned} ^{75}{P}_2\end{aligned}

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

3. Evaluate permutation
\begin{aligned}
^5{P}_5
\end{aligned}

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

5. Evaluate permutation equation
\begin{aligned} ^{75}{P}_2\end{aligned}