Question Detail

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

1. Evaluate combination
\begin{aligned}
^{100}{C}_{97} = \frac{100!}{(97)!(3)!} \\
\end{aligned}

161700
151700
141700
131700

Answer: Option A

Explanation:

\begin{aligned}
^{n}{C}_r = \frac{n!}{(r)!(n-r)!} \\
^{100}{C}_{97} = \frac{100!}{(97)!(3)!} \\
= \frac{100*99*98*97!}{(97)!(3)!} \\
= \frac{100*99*98}{3*2*1} \\
= \frac{100*99*98}{3*2*1} \\
= 161700
\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 ways can the letters of the CHEATER be arranged

20160
2520
360
80

Answer: Option B

Explanation:

As we can see the letter "E" is twice in given word, so Required Number
\begin{aligned}
= \frac{7!}{2!} \\
= \frac{7*6*5*4*3*2!}{2!} \\
= 2520
\end{aligned}

4. 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}

5. 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

Question Detail

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 ?

1. Evaluate combination \begin{aligned} ^{100}{C}_{97} = \frac{100!}{(97)!(3)!} \\ \end{aligned}

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

3. In how many ways can the letters of the CHEATER be arranged

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

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

1. Evaluate combination
\begin{aligned}
^{100}{C}_{97} = \frac{100!}{(97)!(3)!} \\
\end{aligned}

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

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