In how many different ways, the letters of the work ALGEBRA...

In how many different ways, the letters of the work 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

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Key Concepts

Complementary Counting

This counting strategy involves finding the total number of possible arrangements without restrictions and subtracting the number of arrangements that violate the given condition. It is especially useful when it is easier to count the cases that do not meet the criteria rather than those that do.

Permutation with Repeated Elements

This concept is used to calculate the number of arrangements when some elements are identical. In problems like these, the total number of permutations is found by dividing the factorial of the total number of items by the factorial of the number of times each repeated element occurs, which adjusts for overcounting due to indistinguishable arrangements.

Grouping Method

When a problem specifies that certain elements must be together, those elements can be grouped as a single unit. This reduces the total number of items to be arranged. After arranging the groups, any internal arrangements (if applicable) within the group are then considered separately.

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