Question

(i) How many words can be formed with the letters of the word, 'HARYANA'? How many of these

(ii) have H and N together ?

(iii) begin with H and end with N ?

(iv) have 3 vowels together?

Answer 1

(i) The given word ' HARYANA' consists of 7 letters, out of which there are 1 H, 3 A's, 1 R, 1 Y and 1 N.

Total number of words formed by all the letters of the given word = $\frac{7!}{3!}=840.$

(ii) Let us consider as a single letter.

Now, + ARYAA will give us 6 letters out of which there are 3 A' s, 1 R, 1 Y and 1 .

Total number of all such arrangments = $\frac{6!}{3!}=120.$

But, H and N can be arranged amost themselves in $2!$ ways.

Hence, the number of words having H and N together = $(120\times 2)=240.$

(iii) After fixing H in first place and N in last place, we have 5 letters, out of which there are 3 A' s, 1 R and 1 Y.

Hence, the number of words beginning with H and ending with $N=\frac{5!}{3!}=20.$

(iv) The given word contains 3 vowels AAA and let us treat as 1 letter.

Now, we have to arrange 5 letters HRYN+ at 5 places.

Hence, total number of words formed having all vowels together = $5!=(5\times 4\times 3\times 2\times 1)=120.$