# Letter arrangement

• Oct 30th 2012, 09:58 AM
Letter arrangement
onsider
A B C
D
E F G
H
I J K
Here each letter represent a box (identical)( there are 5 rows and 11 boxes) .
Q in how many ways we can put put the alphabets of word LETTER so that no row remain empty.each box contain not more than one alphabet.
any view
• Oct 30th 2012, 11:37 AM
Soroban
Re: Letter arrangement

Quote:

We have:

. . . $\begin{array}{c}\square\!\square\!\square \\[-2mm] \square\quad\; \\[-2mm] \square\!\square\!\square \\[-2mm] \square \quad\; \\[-2mm] \square\!\square\!\square \end{array}$

In how many ways we can put the letters of the word LETTER in the boxes,
so that no row is empty and each box contains not more than one letter.

In the first row, place a letter.
There are 6 choices for the letter and 3 choices for its position.
. . There are $6\cdot3\,=\18$ ways.

In the second row, place a letter.
There are 5 choices for the letter; 1 choice for its position.
. . There are $5\cdot1\,=\,5$ ways.

In the third row, place a letter.
There are 4 choices for the letter; 3 choices for its position.
. . There are $4\cdot3 \,=\,12$ ways.

In the fourth row, place a letter.
There are 3 choices for the letter; 1 choice for its position.
. . There are $3\cdot1 \,=\3$ ways.

In the fifth row, place a letter.
There are 2 choices for the letter, 3 choices for its position.
. . There are $2\cdot3 \,=\,6$ ways.

So, there are: . $18\cdot5\cdot12\cdot3\cdot6 \:=\:19,\!400$ ways
. . to place one letter in each row.

For the last letter, there 6 choices of boxes.

Hence, there are: . $19,\!400\cdot6 \:=\:116,\!640$ ways
. . to place the six letters in the boxes.

But there is duplication caused by
. . the two identical L's and two identical E's.

Therefore, there are: . $\frac{116,\!640}{2!\,2!} \:=\:29,\!160$ ways
. . to place the letters of the word LETTER in the boxes.