In how many ways a word NEWSPAPER can be arranged

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We have to arrange all the letters of word ARRANGE such that AA and RR come together.

So, Put both A's and R's together,
i.e. consider AA and RR as single entities.

n items can be arranged in n! ways: Proof:

$1^{st}$ item has n options, $2^{nd}$ item has (n-1) options,....$n^{th}$ item has 1 option.

So, total ways are:$$n\cdot(n-1)\cdot(n-2)\cdot\cdot\cdot1$$ i.e. n! ways.

You have AA,RR,N,G,E i.e. 5 entities to be arranged.
You can arrange 5 items amongst themselves in 5! ways.

So, final answer is 5!=120 ways.

The strongest acid amongst the following compounds is:

A. CH3COOH

B. HCOOH

C. CH3CH2CH(Cl)CO2H

D. ClCH2CH2CH2COOH

A. CH3COOH

B. HCOOH

C. CH3CH2CH(Cl)CO2H

D. ClCH2CH2CH2COOH

A. CH3COOH

B. HCOOH

C. CH3CH2CH(Cl)CO2H

D. ClCH2CH2CH2COOH

Exercise :: Permutation and Combination - General Questions

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13.

In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together?

A. 10080 B. 4989600 C. 120960 D. None of these Answer: Option C Explanation: In the word 'MATHEMATICS', we treat the vowels AEAI as one letter. Thus, we have MTHMTCS (AEAI). Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different. Number of ways of arranging these letters = 8! = 10080. (2!)(2!) Now, AEAI has 4 letters in which A occurs 2 times and the rest are different. Number of ways of arranging these letters = 4! = 12. 2! Required number of words = (10080 x 12) = 120960.

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Exercise :: Permutation and Combination - General Questions

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7.

How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?

Answer: Option D Explanation: Since each desired number is divisible by 5, so we must have 5 at the unit place. So, there is 1 way of doing it. The tens place can now be filled by any of the remaining 5 digits (2, 3, 6, 7, 9). So, there are 5 ways of filling the tens place. The hundreds place can now be filled by any of the remaining 4 digits. So, there are 4 ways of filling it. Required number of numbers = (1 x 5 x 4) = 20.

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