How many distinguishable permutations of letters are possible in the word COLORADO?
is it 8!/3!?
Yes, you are correctly dividing by 3! for the three O's
Well, that's not quite right. The word COLORADO has 8 letters, but since some of them are repeated, we can't just use 8! (8 factorial) to find the number of distinguishable permutations.
To correctly calculate it, we need to account for the repeated letters. In this case, we have 3 Os and 2 As. So the formula is 8!/(3!2!).
And when you calculate that, you'll get the correct answer. But hey, all those calculations can make your brain feel like it's doing the Cha-Cha Slide!
To find the number of distinguishable permutations of the word COLORADO, we can use the concept of permutations with repeated elements.
The word COLORADO has 8 letters in total, with two occurrences of the letter O. So, we have 3 repeated letters (O) and 5 distinct letters (C, L, R, A, D).
To calculate the number of distinguishable permutations, we use the formula for permutations with repeated elements:
P = (n!)/(r1! * r2! * ... * rk!)
Where n is the total number of items, and r1, r2, ..., rk are the repeated frequencies of those items.
Plugging in the values:
P = (8!)/(2! * 1! * 1! * 1! * 1! * 2!)
Simplifying:
P = (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (2 * 1 * 1 * 1 * 1 * 1 * 2)
P = (8 * 7 * 6 * 5 * 4 * 3) / (2)
P = 20160 / 2
P = 10080
Therefore, there are 10,080 distinguishable permutations possible for the word COLORADO.
To find the number of distinguishable permutations of the word "COLORADO," we need to consider the total number of letters and account for any repeated letters.
The word "COLORADO" contains 8 letters. However, there are 3 O's that appear twice. To find the number of distinguishable permutations, we can divide the total number of permutations by the factorial of the repeated letters. In this case, we divide by 2! (factorial of 2 for the 2 repeated O's).
Using the formula for permutations, which is n!/(r1! x r2! x ... x rk!), where n is the total number of items and r1, r2, ..., rk are the repeated items, we can calculate the answer:
8! / (2!)
Simplifying further:
8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 / (2 x 1)
This equals:
40320 / 2
Therefore, there are 20160 distinguishable permutations of the word "COLORADO."