0

Three non-replaced cards are randomly selected from a deck containing $3$ red, $3$ blue, $3$ green, and $3$ black cards. Specify a sample space for this experiment

I think there are $220$ combinations.

So I have to write the sample space with $220$ ways? Or there is another way to write it?

user
  • 26,272
Olga Gonzalez
  • 185
  • 1
    Why do you think it is 220 ? (Ie express it as $\binom{12}3$ because ....) – Graham Kemp Apr 19 '21 at 23:45
  • because the are combinations of n elements taken from k to k so its $\frac{n!}{k!*(n-k)!}$ – Olga Gonzalez Apr 19 '21 at 23:46
  • 2
    Depends on if RRB is the same as RBR or if they are different. Also, I think the red balls are indistinguishable, So $R_1 B_1 B_1$ is certainly the same as $R_2 B_1 B_3.$ Otherwise the colours wouldn’t have been a part of the question. – Adam Rubinson Apr 19 '21 at 23:47

1 Answers1

1

If cards for the same color are indistinguishable, we can break down the card selection into three different cases:

  1. Three cards are from the same color: $C_1^4=4$ ways.
  2. Two card are the same color, the third one is from a different color: $C_1^4C_1^3=12$ ways.
  3. Three cards are from different colors: $C_3^4=4$ ways.

Total sample space:4+12+4=20.

If the cards for the same color are distinguishable, the total sample space is $C_3^{12}=220$ as expected in the question.

Star Bright
  • 2,338