I am a tomato ^_^

Probability Level 5

Out of all the permutations of the letters of the word "tomatoes" , the permutations in which 2nd O is the 5th letter of the word, are $\dfrac{a}{b}$ times of total. (The ratio of number of these special permutations to the number of all permutations is $\dfrac{a}{b}$ ) where $\mathrm{gcd(a,b)}= 1$ . Find the value of $a+b$

$\bullet$ This problem is a part of the set Vegetable combinatorics

The answer is 8.

1 solution

Ivan Koswara
Jul 24, 2014

When we fix the positions of the two O's, the remaining letters have a constant number of ways to fill the rest of the spots, so we don't care about the remaining letters; simply count the number of ways where the two O's satisfy the given condition, divided by the number of ways to put two O's.

If the second O is at the fifth position, there are $4$ ways for the first O to go. Meanwhile, in general, there are $\binom{8}{2} = 28$ ways for two O's to go in eight positions. Thus the ratio of satisfying permutations to the total number of permutations is $\frac{4}{28} = \boxed{\frac{1}{7}}$ .

1.do we have to consider 2 O's as distinct or identical if they are distinct then total permutations 8!/2! and keeping 1 of the O at the 5th place remaining 7 letters can permuted in 7!/2! so a/b=7!/8!=1/8 a+b=9?
2. and if we consider both O' s to be identical then , total permutations=8!/2! * 2! and required permutations=7!2! a/b=1/4 a+b=5? what is the mistake in the solution?

Cody Martin - 6 years, 9 months ago

Even I'm stuck there

Mayank Singh - 6 years, 3 months ago

The answer is independent of that.

Calvin Lin Staff - 5 years, 6 months ago

Note: Tomatoes are fruits, not vegetables. =="

Samuraiwarm Tsunayoshi - 6 years, 10 months ago

And this is a math website, not botany. And that's true only in theory, not in practise.

Nicolas Bryenton - 6 years, 10 months ago

I am a tomato ^_^

The answer is 8.

1 solution

0 pending reports