Rom

Thank you.  I don't generally use generating functions to answer these sorts of questions because

if the OP doesn't understand the counting method, generating functions just seem to come from

outer space and magically provide answers.

I can see that (for example) they treat stat and atst as two different combinations.

They aren't.

They are both the combination astt.

If no ordering matters at all then 34 is the correct answer.

I'm not sure what ordering they consider distinct to come up with 125

Is "nosceince" supposed to be "nonsense" ?

possibly 1, but probably not

definitely 2

definitely 3

not 4

\(\text{This one is the reverse of the others}\\ \text{Let }w \text{ be the weight you're after in this problem}\\ 0.2 = \Phi\left(\dfrac{261-w}{9.4}\right)\)

\(\Phi^{-1}(0.2) = \dfrac{261-w}{9.4}\\ \text{So use your table or software or whatever to look up the argument, }x,\\ \text{to }\Phi(x) \text{ that corresponds to }\Phi(x) = 0.2\\ \text{Once you've got that value simple algebra will solve for }w\)

\(\text{similar to last one}\\ p = \Phi\left(\dfrac{37-33}{2}\right)-\Phi\left(\dfrac{29-33}{2}\right)\\ n = 400 p\)

\(\text{Let }\Phi(x) \text{ be the CDF of the standard normal distribution}\\ P[\text{weight > 165}] = 1 - P[\text{weight < 165}] = \\ 1 - \Phi\left(\dfrac{165-176}{11.4}\right)\approx \\ 1 - \Phi(-0.964912) =p \text{ (meaning you figure out what }p \text{ is)}\)

\(p\cdot 120 = ?\\ \text{match what you get above with the choice list}\)

\(2,3,\dots, 10 = \{2^1, 3^1, 2^2, 5^1, 2^1 \cdot 3^1,7^1, 2^3, 3^2, 2^1\cdot 5^1\}\\ \text{Using the max exponents for common factors we obtain}\\ LCM = 2^3 \cdot 3^2 \cdot 5^1 \cdot 7^1 = 2520\)

\(\text{Let }x= 2t+3\\ t = \dfrac{x-3}{2}\\ y = 3-3t = 3 - 3\left(\dfrac{x-3}{2}\right)\\ y = 3 -\dfrac 3 2 x+\dfrac 9 2 \\ y = -\dfrac 3 2 x + \dfrac{15}{2}\)

I'll let you work through what you need to verify.  It should be trivial given the formula for the line.

For the first part, pick any t and show that the resulting x and y obey the last formula above.

For the second part pick any x and calculate the associated y using that formula.

Then show that the a single value of t produces them.