Ray and Lilli are designing their own shoes. They have 4 color options for the base and 6 color options for the accent. They can also pick between 2 fonts for the embroidery. What is the probability that Ray and Lilli design identical shoes?

Remark: in the solution, the word Combination is used in the daily usage of the word, not in the combinatorial (mathematical) language.

Let the available 4 base colors be {b_1, b_2, b_3, b_4},

the available 6 accent colors be   {a_1, a_2, a_3, a_4, a_5, a_6},

and the available 2 fonts for the embroidery be {e_1, e_2},

so we have 4 b's, 6a's and 2 e's.

The event that Ray and Lilly design identical shoes can happen in the following way:

Ray has in total 4*6*2=48 possible combinations, and in each of these combinations, Lilly makes the same exact combinations of the 3- b's, a's and  e's.

The sample space is 48 possible combinations of Ray * 48 possible combinations of Lilly = 48*48
(among these there are 48 matches of the combinations)

Thus P(designing identical shoes)=48/(48*48)=1/48=0.02


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