Question

A word is to be formed using some or all the 8 letters.
P A N D E M I C
Find the total number of ways of forming the word if
a) The word is 8-letter long. The letters can be in any combinations. (3 marks)
b) The word is 7-letter long. The word must begin with a consonant. (5 marks)
c) The word is 2-letter long. (3 marks)
d) The word is 6-letter long. Both A and I must be chosen and placed next to one another. (7 marks)
e) The word is 9-letter long. State a reason for your answer.

Answer 1

Given word is

PANDEMIC

which has 8 letters which are all different.

a) Total number of ways of forming 8 letters long word in any combinations will be

8! = 40320 ways.

As we have 8 different letters so the first letter of the 8 letters long word can be selected in 8 ways.

Next, we will have 7 different letters so the second letter of the 8 letters long word can be selected in 7 ways.

In a similar way, the other 6 letters can be selected in 6, 5, 4, 3, 2, and 1 ways respectively.

So total ways will be $8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 8!$ ways

b) In the word PANDEMIC we have three vowels which are A, E, I, and five consonants which are P, N, D, M, C.

To form 7 letter long word which will begin with a consonant the beginning letter can be chosen in 5 ways as we have five consonants.

Next, we will have 7 different letters so the second letter of the 7 letters long word can be selected in 7 ways.

In a similar way, the other 5 letters can be selected in 6, 5, 4, 3, and 2 ways respectively.

So, the total ways will be $5 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 = 5 \times 7!$ ways.

So, the required no of ways will be $5\times 7! = 25200$ ways.

c) To form 2 letter long word

We have 8 different letters so the first letter of the 2 letters long word can be selected in 8 ways.

Next, we will have 7 different letters so the second letter of the 2 letters long word can be selected in 7 ways.

So, the total ways will be $8 \times 7 = 56$ ways.

So, the required no of ways will be 56 ways.

d) To form 6 letter long word such that both A and I must be chosen and placed next to one another.

Now the ways in which we can place A and I in the 6 letters long word such that they are placed next to one another will be 5 ways.

Also, the two letters A and I can be arranged in 2 ways such that they are placed next to one another.

After this, we will have four letters to choose from with 6 letters remaining which can be done in 6, 4, 5, and 3 ways.

So, the total ways for the four letters will be $6 \times 5 \times 4 \times 3 = 360$ ways.

Now the total ways in which we can place A and I in the 6 letters long word such that they are placed next to one another will be 2 times 5 = 10 ways.

So, the required no of ways will be $360\times 10 = 3600$ ways.

e) The no of ways to form a 9 letter long word will be 0 ways as we have 8 letters to choose, from which we can form is maximum of 8 letter long words.