Consider a bag of balls that has 100 red, 100 yellow, and 100 green balls a) How

Question

Consider a bag of balls that has 100 red, 100 yellow, and 100 green balls a) How many color combinations can you get by drawing 10 balls from the bag? b) How many color sequences can you get by drawing 5 balls from the balls? c) How many color combinations of 10 balls have EXACTLY two yellow balls? d) How many color combinations of 10 balls have AT LEAST two yellow balls?

Explanation / Answer

a. Drawing 10 balls : x1 +x2 +x3 = 10 (where x1 = # of red balls, x2 = # of yellow balls etc). Where xi>=0. The number of solutions of this are 12C2 -> (3+10-1)C(3-1). This is done by a star and bars method - among 10 stars, place 2 bars, so that we get 3 numbers that are number of red, yellow and green balls. So answer is 12C2. b. Color sequences of 5 balls - here we select one of the 3 colors for first place, again one of the 3 for second place ... till the fifth place. So, #ways = 3^5 possible color sequences. c. how many combinations with 2 yellow -> so we just need to select # green and red balls -> x1+x2=8 -> # possiblilites = (8+2-1)C(2-1) = 9C1. d.atleast 2 yellow -> x1 + x2 + x3 = 10 where x1 >=2. put y1 = x1-2. So, y1 + x2 + x3 = 8 (substitute.) Now use the same solution. So, # of ways = (8+3-1)C(3-1) = 10C2. Hence proved. Message me if you have any doubts.

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