The first letter we have to choose is M which is our only option. This gives us a probabililty of 1/10.
Next, the only letter where we can get our A is from the second letter. Again, this gives a probability of 1/10.
When choosing the letter T, however, we have to consider the fact that there are two T's in MATHCOUNTS. This gives a probability of 1/5 of choosing it next.
Again, there is only one H so there is 1/10 probability
Multiplying all of these gives us: 1/5000.
Which I think is somehow wrong, someone please correct me.
Use Simon's Favorite Factoring trick!
ab + 2a + 2b = -16 becomes a(b + 2) + 2b = - 16.
Adding 4 on each side gives us:
a(b + 2) + 2(b + 2) = -12, simplifying to (a + 2)(b + 2) = -12
Doing this factoring method for other equations gives us:
(a + 2)(b + 2) = -12
(a + 2)(c + 2) = 30
(b + 2)(c + 2) = -40.
Multiplying those equations and square rooting gives us:
(a + 2)(b + 2)(c + 2) = 120
Applying thiss gives us
120/(c + 2) = - 12
120/(b + 2) = 30
120/(a + 2) = - 40
Can you solve for a, b, c from here?