Looking for help with problem 22 How many 5bit strings begin

Looking for help with problem 22?

How many 5-bit strings begin with 1, 11, 111, 1111 or 11111? The first day on the job for a college graduate is January 2. He owns 4 sports jackets, 8 pairs of dress pants and 5 ties. Any jacket, pair of pants and tie can be worn together. If he wear a different combination to work each day (weekday), then what is the earliest month that be will need to buy new clothes to avoid the same combination of clothes that he wore earlier? A bowl contains 18 blue marbles, 15 red marbles, 15 green marbles and 12 yellow marbles. What is the minimum number of marbles that must be removed from the bowl to be guaranteed that the number of marbles selected of some color is at least two more than the number of marbles of some other color?

Solution

22. The bowl contains 18 blue, 15 red, 15 green and 12 yellow marbles.

The objective is to remove minimum number of marbles to ensure that the number of marbles selected of some color is at least two more than the number of marbles of some other color.

Therefore, the minimum difference between two selected color is two.

So, under the given condition, it is ideal to have 17 blue, 15 red, 13 green and 11 yellow marbles.

11 < 13 < 15 < 17, and the difference is 2 between the consecutive numbers.

Therefore, (18 – 17) = 1 blue, (15 – 13) = 2 green and (12 – 11) = 1 yellow marble must be removed from the bowl.

In other words, the minimum number of marbles the must be removed from the bowl to ensure that the number of marbles selected of some color is at least two more than the number of marbles of some other color, is equal to (1 + 2 + 1) = 4.

(Answer)