Answer: ![\dfrac{13}{2660}](https://tex.z-dn.net/?f=%5Cdfrac%7B13%7D%7B2660%7D)
Step-by-step explanation:
Given : A snack-size bag of M&Ms candies is opened. Inside, there are 12 red candies, 12 blue, 7 green, 13 brown, 3 orange, and 10 yellow.
Total candies = 12+12+7+13+3+10=57
Three candies are pulled from the bag in succession, without replacement.
Number of combinations of selecting r things out of n things = ![^nC_r=\dfrac{n!}{r!(n-r)!}](https://tex.z-dn.net/?f=%5EnC_r%3D%5Cdfrac%7Bn%21%7D%7Br%21%28n-r%29%21%7D)
Number of combinations of selecting 3 things out of 57 things
= ![^{57}C_3=\dfrac{57!}{3!(57-3)!}\\\\=\dfrac{57\times56\times55\times54!}{(6)(4)!}\\\\=57\times56\times55=175560](https://tex.z-dn.net/?f=%5E%7B57%7DC_3%3D%5Cdfrac%7B57%21%7D%7B3%21%2857-3%29%21%7D%5C%5C%5C%5C%3D%5Cdfrac%7B57%5Ctimes56%5Ctimes55%5Ctimes54%21%7D%7B%286%29%284%29%21%7D%5C%5C%5C%5C%3D57%5Ctimes56%5Ctimes55%3D175560)
i.e. Total outcomes = 175560
Number of combination of selecting 2 blue candies out of 12
=![^{12}C_2=\dfrac{12!}{2!(12-2)!}\\\\=\dfrac{12\times11\times10!}{2\times10!}=66](https://tex.z-dn.net/?f=%5E%7B12%7DC_2%3D%5Cdfrac%7B12%21%7D%7B2%21%2812-2%29%21%7D%5C%5C%5C%5C%3D%5Cdfrac%7B12%5Ctimes11%5Ctimes10%21%7D%7B2%5Ctimes10%21%7D%3D66)
Number of combination of selecting 1 brown candies out of 13
=![^{13}C_1=\dfrac{13!}{1!(13-1)!}\\\\=\dfrac{13\times12!}{1\times12!}=13](https://tex.z-dn.net/?f=%5E%7B13%7DC_1%3D%5Cdfrac%7B13%21%7D%7B1%21%2813-1%29%21%7D%5C%5C%5C%5C%3D%5Cdfrac%7B13%5Ctimes12%21%7D%7B1%5Ctimes12%21%7D%3D13)
Favorable outcomes = No. of combination of selecting 2 blue candies x No. of combination of selecting 1 brown candies
= 66 x 13 = 858
The probability that the first two candies drawn are blue and the third is brown =![\dfrac{\text{Favorable outcomes}}{\text{Total outcomes}}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Ctext%7BFavorable%20outcomes%7D%7D%7B%5Ctext%7BTotal%20outcomes%7D%7D)
![=\dfrac{858}{175560}\\\\=\dfrac{13}{2660}](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B858%7D%7B175560%7D%5C%5C%5C%5C%3D%5Cdfrac%7B13%7D%7B2660%7D)