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1. You have four tiles that say M, A, T, and H. How many words can you form from these tiles? For example, you can form "AMH" and "TH". (The words do not have to be valid English words.)

 

2. How many numbers among 1, 2, 3,  1000 are not divisible by 9?

 

3. How many ways can a domino be placed on a  chessboard? Each "half" of the domino must cover exactly one square of the chessboard. An example is shown below.
https://latex.artofproblemsolving.com/e/1/8/e18b44e5bbf7273f8954138f893742b9742ac162.png

 

4.andrew chooses a number from 1 to 100, and Mary also chooses a number from 1 to 100. (They may choose the same number.) It turns out that the product of their numbers is even. In how many ways could Andrew and Mary have chosen their numbers?

 

5.How many 5-digit numbers have at least one zero?

 

6.Dmitri has a pair of standard dice; one die is blue, and the other die is yellow. He rolls both of his dice. How many ways could the number on the blue die be larger than the number on the yellow die?

 Apr 10, 2019
 #1
avatar+4299 
+3

For, problem 2, I suppose you are including the numbers in between.

 

I suggest you using a strategy called "complementary counting," subtracting the unwanted from the total.

 

We can first check the multiples of nine are between 1 and 1000, which is from 9 all the way up till 999, and that is 111 integers.

 

There are a total of 1000 numbers, and the answer should be \(1000-111=\boxed{889}\) integers.

 Apr 10, 2019
 #2
avatar+4299 
+2

3) You can have (Even, Even), (Even, Odd), (Odd, Even). The last (Odd, Odd) will result in odd no matter what...

 

So, we have 50*50*3=7500 ways?
 

 Apr 11, 2019
 #6
avatar+4299 
0

Meant 4).

tertre  Apr 11, 2019
 #9
avatar+102320 
0

Good job, tertre   !!!

 

Correct !!!

 

 

cool cool  cool

CPhill  Apr 11, 2019
 #3
avatar+102320 
+2

6.Dmitri has a pair of standard dice; one die is blue, and the other die is yellow. He rolls both of his dice. How many ways could the number on the blue die be larger than the number on the yellow die?

 

We have a total of  6 * 6  =  36 outcomes

 

6 outcomes will be the same   ....i.e., (1, 1) (2, 2)  , etc.

 

15 will have the number on the yellow die > the number on the blue die

 

And the same number of outcomes will have the number on the blue die > the number on the yellow die

 

 

cool cool cool

 Apr 11, 2019
 #4
avatar+18754 
+3

I will assume you can have 1 letter words like   'M'  'A'  'T'  and 'H'  

4 p 1  +   4 p 2 +  4 p 3  +  4 p 4=

4                12        24           24        = 64 'words'

 Apr 11, 2019
 #5
avatar+18754 
+2

#3    I count  24 ways if the domino halves are indistiguishable

 Apr 11, 2019
 #7
avatar+1013 
-4

Great Job C-Phill and guys 

64 words 

Nickolas  Apr 11, 2019
edited by Nickolas  Apr 11, 2019

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