Before you go any further, I would like you to know that I'm not the best at math, so my answers might be wrong.
a) If it starts at 1 and goes to 99, we know that the answer is 99 numbers. (Even you should be able to figure this one out).
Wait wait wait, but if we count each digit seperately, then it would actually be 189 digits (look below for how)
b) I have two answers for this, if you mean that one of the digits as in we count 10 as a digit, then there would be no probability for Sam to pick a 0, because there's no 0's on the list.
The other option is we count 10 as a 1 and a 0. In this case we have 10, 20, 30 to 90, and thus we have 9 0's. The total number of digits is:
1-9 (9 digits)
10-99 (180 digits)
Total: 189 digits
And we have 9/189 = 1/21
c) The sum of all the digits hmmmmm. Again, I don't know if digits mean each integer seperately or to split an integer. But, if it's the first one, then there are 99(99+1)/2 = 50 * 99= 4950 digits ( I tink).
The other option is splitting each integer, and for this, there's an easy way we can do this. (I think).
First we note how many 1's we have (1,10,11,12,13,14,15,16,17,18,19,21,31...91) Altogether, that is 19 1's (Check that if I counted wrong). So that is 19 of each number. So we have the following:
19 * 1 + 19 * 2 + 19 * 3 + 19 * 4 + 19 * 5 + 19 * 6 + 19 * 7 + 19 * 8 + 19 * 9 = 855
Hope those answers are correct, be sure to tell me if they're wrong...
We know a rectangle has two congruent pairs of sides.
So, we are trying to find what multiplies to 20 and adds to 9 (half of 18 because we are only counting one pair).
x*y = 20
x + y = 9
The factors of 20 are 1,2,4,5,10,20 and two of those add up to 9.
When we look carefully we find those two are 4 and 5.
Sooo the dimensions of the rectangle are 4 and 5.
(Tell me if anything is wrong here)
I'm gonna take a guess here and say that your diagram meant to look like this:
This can be done with a simple counting principle:
( I actually don't know how to do this that well so I'm taking a wild guess)
From the first to the second we have 3 possible ways.
From the second to the third we also have 3 possible ways.
From the third to the fourth we again have 3 possible ways.
SO, my wild guess says the answer is 27 (but it was a wild guess so check it.