In how many different ways can the letters of | vedclass

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(A) The word $ALGEBRA$ consists of $7$ letters: $A, L, G, E, B, R, A$. Here,$A$ appears $2$ times,and $L, G, E, B, R$ appear $1$ time each.$(I)$ When

Vedclass · Accepted Answer

$(I)$ $720$,$(II)$ $1800$

Vedclass · Answer

(A) The word $ALGEBRA$ consists of $7$ letters: $A, L, G, E, B, R, A$. Here,$A$ appears $2$ times,and $L, G, E, B, R$ appear $1$ time each.$(I)$ When the $2$ $A$s are together,we treat the pair $(AA)$ as a single unit. Now we have $6$ units to arrange: $(AA), L, G, E, B, R$. The number of ways to arrange these $6$ units is $6! = 720$.$(II)$ To find the number of ways where the $2$ $A$s are not together,we subtract the number of arrangements where they are together from the total number of arrangements.Total arrangements = $\frac{7!}{2!} = \frac{5040}{2} = 2520$.Number of ways where $A$s are not together = $	ext{Total} - 	ext{Together} = 2520 - 720 = 1800$.

In how many different ways can the letters of the word $ALGEBRA$ be arranged in a row if:
$(I)$ The $2$ $A$s are together?
$(II)$ The $2$ $A$s are not together?

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In how many different ways can the letters of the word $ALGEBRA$ be arranged in a row if:$(I)$ The $2$ $A$s are together?$(II)$ The $2$ $A$s are not together?