Every US coin is stamped with the year in which it was minte

Every US coin is stamped with the year in which it was minted. How many coins do you need to have in your pocket to be assured that at least 2 of them have the same last digit ? How many do you need to be assured that at least 2 of them have the same first digit?

Solution

1) There is a possibility that 10 coins have different last digit but the 11th coin will coincide with the last digit from any of the 10 coins . Hence , minimum number of coins required are 11 coins

2) For first digit , we can have 1xxx type or 2xxx type but the third coin has to be among the these two type . Hence , there is need fot 3 coins to have such combination .