Question

How many different strings can be made from the letters in AARDVARK, using all the letters, if all three As must be consecutive?

Short answer

The total number of 360 strings can be made from the letters AARDVARK

Step-by-step solution

Step 1: Use the formula for string factorial

Total number of arrangements of n objects if al are different = n!

If r! of them are same number of ways are given by

\(\begin{array}{l}\frac{{n!}}{{{n_1}!{n_2}!..{n_k}!}} = \frac{{n!}}{{r!}}\\r! = {n_1}!{n_2}!..{n_k}!\end{array}\)

The n! Is string length

r! are outcome of strings

Step 2: Solution of the number of strings made from the letters AARDVARK

To count the number of permutations of letters of AARDVARK with the 3 A's consecutive.

we can treat the 3 consecutive A's as one letter, i.e., count the number of permutations of 2 R's (indistinguishable from each other), 1 D, 1 V, 1 K, and 1 "AAA", for a total of 6 "letters".

Here, applied the factorial function

\(\begin{array}{l}n! = 6!\\r! = 2!,1!,1!,1!,1!\\\frac{{n!}}{{r!}} = \frac{{6!}}{{2!.1!.1!.1!.1!}}\\ = 360strings\end{array}\)