Rom

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\(\text{Let }S \text{ be the set of all positive integers invertible mod }4!\\ \text{you can partition }S \text{ into pairs of inverses }(x,~x^{-1})\\ -\\ \text{The product of all the elements mod 4! is equal to the product of the }\\ \text{products of the individual pairs mod 4!}\\ -\\ x \cdot x^{-1} \pmod {4!} = 1 \text{ by definition of multiplicative inverse}\\ \text{The product of all these 1's will be 1, and thus the overall product mod 4!}=1\)

good catch Melody.

terrible use of variables in the OP.

\(P[6] = \dfrac{9}{10}\cdot \dfrac 1 6 + \dfrac{1}{10} \dfrac{18}{100}= \dfrac{21}{125} \)

\(P[\text{biased|6}] = \dfrac{P[\text{6|biased}]P[\text{biased}]}{P[6]} = \\ \large \dfrac{\frac{18}{100}\frac{1}{10}}{\frac{21}{125}} = \dfrac{3}{28}\)

\(P[\text{biased|!6}] = \dfrac{P[\text{!6|biased}]P[\text{biased}]}{P[\text{!6}]}=\\ \dfrac{\frac{82}{100}\frac{1}{10}}{1-\frac{21}{125}} = \dfrac{41}{416}\)

You can do the last one... see if you learned anything

poof

\({7,7,9,10,22}\)

This problem is ill stated.

Unless you've left some info out there are infnite solutions for y in terms of x.

There end up being two free parameters, v0 and theta, the angle at which the throw is made with

respect to the x axis.

The ball could the thrown straight up and satisfy what's given or it could thrown at any angle and 

a v0 could be found to solve the conditions listed.

Did you leave something out?

\(\text{If there are 8 T's there are 8 C's, so we are left with 14 slots}\\ \text{This means that A=G=7}\\ P[\text{8 T's}] = \dbinom{30}{8}\dbinom{22}{8}\dbinom{14}{7}(0.29)^7(0.23)^8(0.27)^7(0.21)^8\)

Now the problem might actually mean there are at least 8 T's.  In which case we'd modify above to be

\(\large \sum \limits_{t=8}^{15}~\dbinom{30}{t}\dbinom{30-t}{t}\dbinom{30-2t}{15-t}(0.29)^{15-t}(0.23)^t(0.27)^{15-t}(0.21)^t\)

\(24000=(1-0.04)O_{lastyr}\\ O_{lastyr} = \dfrac{24000}{0.96} = 25000\)

3)

\(P[A] = \left(1-\dfrac 1 2\right)\left(\dfrac 1 4\right) = \dfrac 1 8\\ P[B]= \text{.... are you supposed to be able to do this w/o software?}\\ \text{It's next to impossible by hand}\\ \text{unless I'm missing some major trick which is certainly possible}\)