How many arrangements can be made from the word
MATHEMATICS when all are taken at a time?
To calculate the number of arrangements that can be made from the word "MATHEMATICS" when all are taken at a time, we need to consider the number of total letters and their frequencies.
The word "MATHEMATICS" has a total of 11 letters. Let's break down the frequency of each letter:
M - 2 times
A - 2 times
T - 2 times
H - 1 time
E - 1 time
I - 1 time
C - 1 time
S - 1 time
To calculate the number of arrangements, we can use the concept of permutations. The formula for permutations is:
nPr = n! / (n - r)!
Where n is the total number of items (letters in this case) and r is the number of items to be selected (in our case, all the letters).
To apply this formula, we substitute the values into the equation:
11P11 = 11! / (11 - 11)!
Now, let's calculate the value:
11P11 = 11! / 0!
Since 0! equals 1, the equation simplifies:
11P11 = 11!
Using the factorial function, we can find the value of 11!:
11! = 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 39,916,800
Therefore, there are 39,916,800 different arrangements that can be made from the word "MATHEMATICS" when all the letters are taken at a time.