Probability.

When three standard dice are tossed, the numbers a,b,c are obtained. Find the probability that a*b*c=180.

 

180=18*10=9*2*5*2=2*2*3*3*5

So one of the numbers has to be 5

the other 2 both have to be 6

6*6*5=180

For ease lets assume that the dice are red green and blue.  Any of thes can be the 5 and the other two have to be 6 so that is 3 ways.

 

How many different combintations are possible.  6*6*6 = 216

 

So the probabily of getting a product of 180  is    3/216   =  1 / 72

The probability is the the coefficient of x^17 in the expansion of the following generating series divided by 6^3.  expand | (x+x^2+x^3+x^4+x^5+x^6)^3

 

x^18+3 x^17+6 x^16+10 x^15+15 x^14+21 x^13+25 x^12+27 x^11+27 x^10+25 x^9+21 x^8+15 x^7+10 x^6+6 x^5+3 x^4+x^3.

 

Since the coefficient of x^17 is 3, therefore the exact probability is: 3 / 6^3=3/216 =1/72 =1.39%.