Permutations and Combinations

Question

Find the number of arrangements using all of the letters in the word LOGARITHMS.

 

How many of these arrangements have the consonatns G and H separated?

 

Ans 2903040

Answer 1

Note that all the alphabets in the word are distinct and that there are 10 in total. Without any constraint there are 10! ways for arranging the letters. If you want G and H separated - work it out as follows - let us place G first, H second and the other letters subsequent to them - such a choice has no impact on the total number of arrangements. If G is to the placed on the leftmost or rightmost site H can only occupy any of 8 other sites (i.e excluding the sites next to where G is placed). If G is placed on a site inbetween that has two neighboring sites, then H can occupy any of 7 sites (excluding the site where G is placed and its two neighboring sites). There is no restriction on the arrangement of the remaining 8 letters. Hence the number of arrangements is: (2*8 + 8*7)*8! = 9!*8 = 2903040

Answer 2

if there were no restrictions there would be 10! ways

10!=10*9*8*7*6*5*4*3*2*1=3,628,800 ways

now subtract the total number of ways that G and H can occur together

these are the ways:

GH345678910

1GH45678910

12GH5678910

123GH678910

1234GH78910

12345GH8910

123456GH910

1234567GH10

12345678GH

now repeat this with HG to get 9 more ways for a total of 18 ways that G and H can occur together

in each case you have used up 2 digits, which leaves 8 digits to be put in the other places

you can place these digits in 8! ways in each of the 18 sequences mentioned

8!*18=8*7*6*5*4*3*2*1*18=725,760 ways

3,628,800-725,760=2,903,040 ways