You have 10 of eah of the typical US coins penny nickel dime

You have 10 of eah of the typical US coins( penny, nickel, dime, quarter) in a bag. If you are choosing coins without looking in the bag, what is the minimum number of coins you must select to guarantee

(a) at least 5 of one type of coin?

(b) at least 5 quarters?

(c) at least 5 quarters and 5 nickels

Solution

(a) at least 5 of one type of coin

P-Penny,N-Nickel,D-Dime and Q-Quarter

Let us suppose that the coin comes in this pattern (P,N,D,Q) after taking 16 coins, we will have in worst case 4 coins of each type and as soon as we take the 17th coin it can be either of the (P,N,D,Q)

if it is P, then it will contain 5P,4N,4D,4Q

if it is N, then it will contain 4P,5N,4D,4Q

if it is D, then it will contain 4P,4N,5D,4Q

if it is Q, then it will contain 4P,4N,4D,5Q

Hence the minimum number of coins required in this case will be equal to 17 coins to guarantee that at least 5 of one type of coin is available in the lot

b) at least five quarters

Let us suppose that the coin comes in this pattern that all Penny comes first then 10 pennies need to be removed in that case

Now let us 10 nicket followed by dime comes, then we have taken 30 coins

Hence we taken 5 more coins which are all quarters to get atleast five quarters

Hence the correct answer is 35

(c) at least 5 quarters and 5 nickels

Let us suppose that the coin comes in this pattern that all Penny comes first then 10 pennies need to be removed in that case

Now let us 10 nickel followed by dime comes, then we have taken 30 coins

Hence we taken 5 more coins which are all quarters to get atleast five quarters (5 nickel have already been taken out in the lot)

Hence the correct answer is 35