NOTE: IS STUDYING \"DISCRETE MATHEMATICS FOR COMPUTER SCIENCE\" ABOUT PERMUATION

Question

NOTE: IS STUDYING "DISCRETE MATHEMATICS FOR COMPUTER SCIENCE" ABOUT PERMUATION AND COMBINATION, AND WANT TO SEE THE WORK OF THE ANSWERS

5/5 pts Question 1 There are 5 red, 3 green, and 8 blue marbles in a box. Each marble is distinct. a) How many ways can 4 marbles be drawn out of the box if all 4 marbles are picked at the same time? 1820 b) How many ways can you pick 4 marbles (all at the same time) where all 4 marbles have the same color? 75 c) How many ways can you pick 4 marbles (all at the same time) where 2 of the marbles are the same color and the other 2 marbles are the same color (but you have two different colors). For instance, you might pick 2 red marbles and 2 green marbles. 394

Explanation / Answer

a)

Total marbles = 5 + 3 + 8 = 16

If 4 marbles are picked at the same time then :

Formula :-

nCr = n!/(r! * (n-r)!) = 16!/(4! * 12!) = 1820 ways

where n = total no of marbles
r = total no of selection

b)
Total marbles = 5 + 3 + 8 = 16

Same color :-

Blue = 8
Therefore,no of way in which marbles picked being blue color = nCr =
Here,
n=8
r=4

Therefore, 8c4 = 8!/(4! * 4!) = 70 ways

Red = 5
Therefore,no of way in which marbles picked being Red color = nCr =
Here,
n=5
r=4

Therefore, 5c4 = 5!/(4! * 1!) = 5 ways

green = 3
Therefore,green marbles are 3 .So , any order if 4 marbles are picked; we can't get all marbles are of same color as green

So,total ways in which 4 marbles picked all being same color is 70 ways + 5 ways = 75 ways

c)
Total marbles = 5 + 3 + 8 = 16

2 marbles of same color :-

Blue = 8
Therefore,no of way in which marbles picked being blue color = nCr =
Here,
n=8
r=2

Therefore, 8c2 = 8!/(2! * 6!) = 28 ways

Red = 5
Therefore,no of way in which marbles picked being Red color = nCr =
Here,
n=5
r=2

Therefore, 5c2 = 5!/(2! * 3!) = 10 ways

green = 3
Therefore,no of way in which marbles picked being green color = nCr =
Here,
n=3
r=2

Therefore, 3c2 = 3!/(2! * 1!) = 3 ways

Combinations:-

1)
green and red :-

3*10 = 30 ways

green and blue :-

3*28 = 84 ways

red and blue :-

10 * 28 = 280 ways

Total ways :-

280 + 84 + 30 ways = 394 ways

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