Final answer:
The number of ways three letters from 'algorithm' can be arranged is 504. For six letters, it is 60,480 ways. If the first letter is 'a', or the first two letters are 'or', there are 6,720 and 840 ways, respectively.
Step-by-step explanation:
The question asks about different permutations of the letters from the word algorithm. In combinatorics, a branch of mathematics, permutations are the different ways in which a set or number of things can be ordered or arranged.
Three-letter permutations
To find the number of ways three of the letters of the word 'algorithm' can be selected and written in a row, we use permutations since the order matters. The formula for permutations is nPr = n! / (n - r)!, where n is the total number of items to choose from, r is the number of items to choose, and ! denotes factorial. Hence, for algorithm which has 9 distinct letters, choosing 3 would be 9P3 = 9! / (9 - 3)! = 9! / 6! = 9 x 8 x 7 = 504 ways.
Six-letter permutations
For six of the letters, it is 9P6 = 9! / (9 - 6)! =
9! / 3! = 9 x 8 x 7 x 6 x 5 x 4 = 60,480 ways.
Six-letters with first letter 'a'
When the first letter must be 'a', we have eight letters left from which to choose the remaining five. Thus, it is 8P5 = 8! / (8 - 5)! = 8! / 3! = 8 x 7 x 6 x 5 x 4 = 6,720 ways.
Six-letters with first two letters 'or'
If the first two letters must be 'or', we then have seven letters left from which to choose the remaining four. So, it is 7P4 = 7! / (7 - 4)! = 7! / 3! = 7 x 6 x 5 x 4 = 840 ways.