Problem:
Kathy has 5 red cards and 5 green cards. She shuffles the 10 cards and lays out 5 of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is nm, where m and n are relatively prime positive integers. Find m+n.
Solution:
Assume without loss of generality that the first card laid out is red. Then the arrangements that satisfy Kathy's requirements are RRRRR, RRRRG, RRRGG, RRGGG, and RGGGG. The probability that Kathy will lay out one of these arrangements is
94⋅83⋅72⋅61+94⋅83⋅72⋅65+94⋅83⋅75⋅64+94⋅85⋅74⋅63+95⋅84⋅73⋅62=12631.
The requested sum is 31+126=157.
OR
Assume without loss of generality that the first card laid out is red. The probability that k of the four remaining laid out cards are red, where 0≤k≤4, is
(94)(4k)(54−k)
Given that there are exactly k red cards, the probability that they are laid out at the start is (k4)1. Hence the required probability is
k=0∑4(94)(54−k)=(94)25−1=12631
The problems on this page are the property of the MAA's American Mathematics Competitions