How many ways can the letter of the word rectangle be arranged such that all the vowels are always together?

In the word 'MATHEMATICS', we'll consider all the vowels AEAI together as one letter.
Thus, we have MTHMTCS (AEAI).
Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice
 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 x 12) =

120960

1) In what ways the letters of the word "RUMOUR" can be arranged?

  1. 180
  2. 150
  3. 200
  4. 230

Answer: D

Answer with the explanation:

The word RUMOUR consists of 6 words in which R and U are repeated twice.
Therefore, the required number of permutations =

How many ways can the letter of the word rectangle be arranged such that all the vowels are always together?

Or,
How many ways can the letter of the word rectangle be arranged such that all the vowels are always together?
= 180

Hence, 180 words can be formed by arranging the word RUMOUR.


2) In what ways the letters of the word "PUZZLE" can be arranged to form the different new words so that the vowels always come together?

  1. 280
  2. 450
  3. 630
  4. 120

Answer: D

Answer with the explanation:

The word PUZZLE has 6 different letters.

As per the question, the vowels should always come together.
Now, let the vowels UE as a single entity.
Therefore, the number of letters is 5 (PZZL = 4 + UE = 1)
Since the total number of letters = 4+1 = 5
So the arrangement would be in 5P5 =

How many ways can the letter of the word rectangle be arranged such that all the vowels are always together?
=
How many ways can the letter of the word rectangle be arranged such that all the vowels are always together?
= 5! = 5*4*3*2*1 = 120 ways.

Note: we know that 0! = 1

Now, the vowels UE can be arranged in 2 different ways, i.e., 2P2 = 2! = 2*1 = 2 ways

Hence, the new words, which can be formed after rearranging the letters = 120 *2 = 240

As we known z is occurring twice in the word ‘PUZZLE’ so we will divide the 240 by 2.

So, the no. of permutation will be = 240/2 = 120


3) In what ways can a group of 6 boys and 2 girls be made out of the total of 7 boys and 3 girls?

  1. 50
  2. 120
  3. 21
  4. 20

Answer: C

Answer with the explanation:

We know that nCr = nC(n-r)

The combination of 6 boys out of 7 and 2 girls out of 3 can be represented as 7C6 + 3C2
Therefore, the required number of ways = 7C6 * 3C2 = 7C(7-6) * 3C(3-2) =

How many ways can the letter of the word rectangle be arranged such that all the vowels are always together?
= 21

Hence, in 21 ways the group of 6 boys and 2 girls can be made.


4) Out of a group of 7 boys and 6 girls, five boys are selected to form a team so that at least 3 boys are there on the team. In how many ways can it be done?

  1. 645
  2. 734
  3. 756
  4. 612

Answer: C

Answer with the explanation:

We may have 5 men only, 4 men and 1 woman, and 3 men and 2 women in the committee.

So, the combination will be

as we know that

nCr=

How many ways can the letter of the word rectangle be arranged such that all the vowels are always together?

So, (7C3 * 6C2) + (7C4 * 6C1) + (7C5)
Or,

How many ways can the letter of the word rectangle be arranged such that all the vowels are always together?
+
How many ways can the letter of the word rectangle be arranged such that all the vowels are always together?
+
How many ways can the letter of the word rectangle be arranged such that all the vowels are always together?

Or, 525 +210+21 = 756

So, there are 756 ways to form a committee.


5) A box contains 2 red balls, 3 black balls, and 4 white balls. Find the number of ways by which 3 balls can be drawn from the box in which at least 1 black ball should be present.

  1. 64
  2. 48
  3. 32
  4. 96

Answer: A

Answer with the explanation:

The possible combination could be (1 black ball and 2 non-black balls), (2 black balls and 1 non- black ball), and (only 3 black balls).

Therefore the required number of combinations = (3C1 * 6C2) + (3C2 * 6C1) + (3C3)
r,

How many ways can the letter of the word rectangle be arranged such that all the vowels are always together?
+
How many ways can the letter of the word rectangle be arranged such that all the vowels are always together?
+
How many ways can the letter of the word rectangle be arranged such that all the vowels are always together?
= 45+18+1 = 64


Permutation and Combination Test Paper 2
Permutation and Combination Concepts

How many ways can the letters of the word rectangle be arranged such that all the vowels are ways together?

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

How many ways can the letters of the word rectangle?

338 words can be made from the letters in the word rectangle.

How many ways can the letters of the word MATHEMATICS be arranged so that the vowels always come together?

∴ Number of ways of arranging these letters =8!/(2!)( 2!) = 10080.

How many ways word arrange can be arranged in which vowels are together?

The number of ways the word TRAINER can be arranged so that the vowels always come together are 360.