Question

how many word can be formed by using all the letters of the word ' ALLHABAD' ?

Answer 1

answer :

the given word contains 9 letters out of which A is repeated 4 times L is repeated 2 times and the rest all are different.

so ,

the requisite number of words = 9! / (4!) (2!) = 7560


be brainly

Answer 2

here is your answer ✌✌

Assuming that a word in this context means any arrangement of letters and not necessarily meaningful.

Initially find out the total number of words that can be formed with starting letter A 

The word ALLAHABAD has 9 letters. There are 4 'A's. Fix one of them in the first position. Since all of them are similar, only one way of doing this.Then there are 8 letters remaining in which 3 'A' and 2 'L's are repeating. These can be arranged in 8!3!.2!8!3!.2! = 3360 ways
Total number of arrangements, starting with A = 3360

Similary, starting with B, see the number of arrangements possible. Fix B at the first position. There are 8 letters remaining in which 4 'A's and 2 'L's are there. Hence, total number arrangements = 8!4!×2!8!4!×2! = 840

Similary, total number of arrangemetns starting with D = 8!4!×2!8!4!×2! = 840

Similary, total number of arrangemetns starting with H = 8!4!×2!8!4!×2! = 840

Hence, 3360 + 840 + 840 + 840 = 5880 words are there before starting the words begining with 'L' (i.e., LAAAABDHL wiill be the 5881th word)

We are now looking at the number of arrangements which can be done with LAA. Fix L at the first position. Fix one 'A' at thte second position. Fix another 'A' at third second position.  There are 6 letters remaining in which 2 'A's are repeating. Total number of arrangemetns = 6!2!6!2! = 360

i.e.,after 5880+360 = 6240 words, words start from LAB AAADHL

In a simialr way, number of words with first letter as LABAA = 4! = 24. 

i.e.,after 6240+24 = 6264 words, words start from LABAD AAHL

Number of words stating with LABADAA = 2! = 2 
Number of words stating with LABADAH = 2! = 2
after 6264+2+2 = 6268 words, words start from LABADALAH