Can someone help me out with these? 1) This problems refers to piles of identica

Question

Can someone help me out with these? 1) This problems refers to piles of identical red, blue and green balls where each pair contains at least 10 balls. how many ways can 10 balls be selected if at most one red balls is selected? 2) a bag contains 20 balls-6 red,6green and 8 purple. We draw 5 balls and at least one is red then replace the. We then draw 5 balls and at most one is green. In how many ways this be done if the balls are considered distinct? 3) 5 distinct cs book, 3 distinct math books and 2 distinct art books are on self. how many ways can theses books be arranged on a shelf if all 5 cs books are on left and both art books are on the right?

Explanation / Answer

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For each answer, give a brief explanation of your reasoning.
There are piles of identical red, blue, and green balls where each pile contains at least 10
balls.
(a) In how many ways can 10 balls be selected if at least one red ball, at least two blue balls,
and at least three green balls must be selected

Combination with repetition. The 3 colors are the categories, and six choices have
already been made (1 red ball, 2 blue balls, 3 green balls), leaving four remaining choices
to be made.

In how many ways can 10 balls be selected if exactly one red ball and at least one blue
ball must be selected?

Combination with repetition. The colors are the categories, but there are only two (blue
and green) since we want exactly one red ball. Two choices have already been made (1
red ball and 1 blue balls), leaving eight remaining choices to be made.

In how many ways can 10 balls be selected if at most one red ball is selected?

If at most one red ball is selected, then either 1 red ball is selected or 0 red balls
are selected. Since these two possibilities are distinct, the sum rule applies with no
overcounting adjustment. In either case, we have two categories, blue and green, and
either 9 or 10 choices of those type of balls.

In how many ways can 10 balls be selected if twice as many red balls as green balls must
be selected?

Four, since the possibilities are (0; 0), (2; 1), (4; 2), and (6; 3), where a pair (r; g) designates
r red and g green balls.

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