Question

In how many different ways, the letters of the word ALGEBRA can be arranged in a row if
(i) The two As are together ?
(ii) The two As are not together ?

• A. (i) 720, (ii) 1800
• B. (i) 620, (ii) 1600
• C. (i) 780, (ii) 1860
• D. (i) 720, (ii) 1600

Answer 1

The correct option is A (i) 720, (ii) 1800

ALGEBRA has seven letters where 2 - A, 1 -L, 1 - G, 1 - E, 1 -B and 1 -R.
(i) Since two A's are always together, we take both the A's as one letter.
If p is the number of arrangements, then
$p=6!=6\times 5\times 4\times 3\times 2\times 1=720.$

(ii) Total number of permutations

$q=\frac{7!}{2!}=\frac{7\times 6\times 5\times 4\times 3\times 2\times 1}{2\times 1}$

= 2520

In these permutations, in some permutations, two A's are together while in the rest they are not together. Hence, the number permutations in which two A's are not together in
q - p = 2520 - 720 = 1800