I'm absolutely terrible at calculating these things and I would like to, especially with SATs coming up, improve my capabilities.
What always gets me is that there are so many types of ways to combine items.
One way might be to figure out rearrangements of a set:
abcd => abcd abdc acbd acdb adbc adcb bacd badc...
…or combinations of the elements of the set used more than once
abcd => aaaa aaab aaac aaad aaba aabb ...
…or combinations limited to two elements
abcd => aa ab ac ad ba bb bc bd ca cb cc...
…or in threes
abcd => aaa aab aac aad aba abb abc abd...
…or ways to combine the elements in pairs
abcd => ab ac ad ba bc bd ca cb cd da...
…or pairs without repeating
abcd => ab ac ad bc bd...
Point taken, I'm sure. I know they're all incredibly simple: multiply two numbers or find the summation and then divide by the number or times it's going to repeat… Again though, I have no idea which calculations to apply to which variations. I know it should be logical, but I can never quite figure it out.
And then there was an SAT practice problem online… http://sat.collegeboard.org/practice/sat-question-of-the-day?questionId=20120221&oq=1
Of 5 employees, 3 are to be assigned an office and 2 are to be assigned a
cubicle. If 3 of the employees are men and 2 are women, and if those
assigned an office are to be chosen at random, what is the probability
that the offices will be assigned to 2 of the men and 1 of the women?
I don't even know where to begin to solve this problem..
Best Answer
There are $\binom{5}{3}$ ways to choose the $3$ people for the offices, all equally likely. There are $\binom{3}{2}\binom{2}{1}$ ways to choose $2$ men from the $3$ available, and $1$ woman from the $2$ available. So the required probability is $$\frac{\binom{3}{2}\binom{2}{1}}{\binom{5}{3}}.$$