Question

How many different strings can be made from the letters in ABRACADABRA, using all the letters?

Short answer

The total number of 83160 strings can be made from the letters ABRACADABRA

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 ABRACADABRA

In "ABRACADABRA", there are 5 A's, 2 B's, 1 C, 1 D, and 2 R's, for a total of 11 letters.

So we are trying to calculate the number of permutations of 11 letters, with the 5 A's indistinguishable from each other, 2 B's indistinguishable from each other, and 2 R's indistinguishable from each other.

Here, applied the factorial function

\[\begin{array}{l}n! = 11!\\r! = 5!,2!,2!\\\frac{{n!}}{{r!}} = \frac{{11!}}{{5!.5!.2!}}\\ = 83160strings\end{array}\]