NCERT Class 11 Mathematics Solutions: Chapter 7 – Permutation and Combinations Miscellaneous Exercise Part 2 (For CBSE, ICSE, IAS, NET, NRA 2023)

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Permutation vs Combinations

1. If the different permutations of all the letter of the word are listed as in a dictionary, how many words are there in this list before the first word starting with ?


In the given word , there are letters out of which, appear times and all the other letters appear only once.

The words that will be listed before the words starting with in a dictionary will be the words that start with only.

So, to get the number of words starting with , the letter is fixed at the extreme left position, and then the remaining letters taken all at a time are rearranged.

Since there are Is and Ns in the remaining letters,

Number of words starting with

So, the required numbers of words is .

2. How many -digit numbers can be formed from the digits which are divisible by and no digit is repeated?


A number is divisible by if its units digits is .

So, is fixed at the units place.

So, there will be as many ways as there are ways of filling vacant places

Five Vacant Places

in succession by the remaining digits .

The vacant places can be filled in ways.

Hence, required number of digit numbers

3. The English alphabet has vowels and consonants. How many words with two different vowels and different consonants can be formed from the alphabet?


different vowels and different consonants are to be selected from the English alphabet.

Since there are vowels in the English alphabet, number of ways of selecting different vowels form the alphabet

Since there are consonants in the English alphabet, number of ways of selecting different consonants from the alphabet .

So, number of combinations of different vowels and different consonants

Each of these combinations has letters, which can be arranged among themselves in ways.

Therefore, required number of words