A polygon 7 sides.How many diagonals can be formed?

Correct Answer: 100

Explanation:

1 million distinct 3 digit initials are needed.

 

Let the number of required alphabets in the language be ?n?.

 

Therefore, using ?n? alphabets we can form n * n * n =  n 3 distinct 3 digit initials.

 

Note distinct initials is different from initials where the digits are different.

 

For instance, AAA and BBB are acceptable combinations in the case of distinct initials while they are not permitted when the digits of the initials need to be different.

 

This  n 3 different initials = 1 million 

i.e.  n 3 = 10 6  (1 million = 10 6)

  => n =  10 2 = 100

 

Hence, the language needs to have a minimum of 100 alphabets to achieve the objective.

Correct Answer: 17280

Explanation:

There are 4 books on fairy tales and they have to be put together. They can be arranged in 4! ways.

 

Similarly, there are 5 novels.They can be arranged in 5! ways.

 

And there are 3 plays.They can be arranged in 3! ways.

 

So, by the counting principle all of them together can be arranged in 4!´5!´3! ways = 17280

Correct Answer: 164

Explanation:

For a triangle, we need 3 non-collinear points. So with 12 points (when all the 12 are such that any three non-collinear is 12 C 3. But among them 8 points are collinear.

 

If all these 8 points are different we get  8 C 3 triangles as they are collinear.

 

In  12 C 3 triangles, we do not get  8 C 3 triangles

 

Therefore, The number of triangles we get = 12 C 3 - 8 C 3 = 164

Correct Answer: 531

Explanation:

Given in the question that, there are 7 boys and 6 girls. 

Team members = 5

Now, required number of ways in which a team of 5 having atleast 3 girls in the team = 

$$\begin{gathered} 6 C 3 x 7 C 2 + 6 C 4 x 7 C 1 + 6 C 5 \\ = 6 x 5 x 4 3 x 2 x 1 x 7 x 6 2 x 1 + 6 x 5 x 4 x 3 4 x 3 x 2 x 1 x 7 + 6 x 5 x 4 x 3 x 2 5 x 4 x 3 x 2 x 1 \\ = 420 + 105 + 6 \\ = 531 . \end{gathered}$$

Correct Answer: 6! x ?P?

Explanation:

We can initially arrange the six boys in 6! ways.
Having done this, now three are seven places and six girls to be arranged. This can be done in ?P? ways.

Hence required number of ways = 6! x ?P?

Correct Answer: 455

Explanation:

There are 15 dots in total,and to make a triangle we need to select any three of those dots.

 

So, 15 C 3 = 455

Correct Answer: 42

Explanation:

The question requires you to find number of the outcomes in which at most 3 coins turn up as heads.

i.e., 0 coins turn heads or 1 coin turns head or 2 coins turn heads or 3 coins turn heads.

 

The number of outcomes in which 0 coins turn heads is  6 C 0=1 

The number of outcomes in which 1 coin turns head is = 6 C 1=6 

The number of outcomes in which 2 coins turn heads is 6 C 2=15 

The number of outcomes in which 3 coins turn heads is 6 C 3=20

 

Therefore, total number of outcomes =1+6+15+20= 42 outcomes

Correct Answer: 70

Explanation:

All the boxes contain distinct number of chocolates.
For each combination of 4 out of 8 boxes, the box with the greatest number has to be given to the first person, the box with the second highest to the second person and so on.

 

The number of ways of giving 4 boxes to the 4 person is: 8 C 4= 70

Correct Answer: 108

Explanation:

The number of words beign with A is 4! 

The number of words beign with E is 4! 

The number of words beign with M is 4! 

The number of words beign with R is 4!

 

Number of words beign with VA is 3! 

Words beign with VE are VEAMR 

VEARM

VEMAR

VEMRA

VERAM

VERMA 

Therefore, The Rank of the word VERMA = 4 x 4! + 3! + 6 = 96 + 6 + 6 =108             

Correct Answer: 5040

Explanation:

The Word LOGARITHMS  contains 10 letters.

To find how many 4 letter words we can form from that =  P 4 10 =10x9x8x7 = 5040.