Use the pigeonhole principle to give solutions to the following problems: (a) How many times must a single die be rolled to guarantee that some number is obtained at least twice? (b) How many times must two dice be rolled to guarantee that the same total score is obtained at least twice? (c) How many times must two dice be rolled to guarantee that the same total score is obtained at least three times?

Let us use the pigeonhole principle to give solutions to the following problems.

(a) Since there are 6 different outcomes of a single die, we conclude that according to the pigeonhole principle a single die must be rolled $6+1=7$ times to guarantee that some number is obtained at least twice.

(b) The total score can be one of the following numbers: $2,3,4,5,6,7,8,9,10,11,12.$ Since there are 11 different outcomes, we conclude that according to the pigeonhole, two dice must be rolled 12 times to guarantee that the same total score is obtained at least twice.

(c) The total score can be one of the following numbers: $2,3,4,5,6,7,8,9,10,11,12.$ Since there are 11 different outcomes, we conclude that according to the pigeonhole, two dice must be rolled $11\cdot 2+1=23$ times to guarantee that the same total score is obtained at least three times.