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Combinatorics Math Challenge II-A 2.25 and 2.29

 
 
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Combinatorics Math Challenge II-A 2.25 and 2.29
by Tina Jin - Saturday, January 1, 2022, 3:56 PM
 

Hello.

For problem 2.25, I know that the # of rectangles should be 7C2 * 6C2, but I don't know how to do the squares. As I looked into the solution on the Quick Response Quiz,

65+54+43+32+21=70

What does this mean? Is it a formula? I'm really confused.


For problem 2.29, should it be  9C4*4! * 8 * 7 ?  4! is the # of ways to take math classes, and 8*7 is the# of ways to pick 2 more classes. and 9 C4 is the # of ways to arrange the math classes.


Thank you

 
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Re: Combinatorics Math Challenge II-A 2.25 and 2.29
by John Lensmire - Friday, January 14, 2022, 1:42 PM
 

Thanks for your patience! (We'll also get to all your number theory questions when we have a chance.)

For 2.25, there isn't really a formula for there squares, but there is a pattern. The idea is to start by counting the 1x1 squares, then the 2x2 squares, etc.

The 1x1 squares should be fairly easy, this is all the squares, so there are 6*5 = 30 in total. For the 2x2 squares we can do a similar idea. Moving horizontally we can create five 2x2 squares and moving vertically we can create four different 2x2 squares. This is how we get 5*4 = 20 in total. This pattern can continue for the 3x3, 4x4, and 5x5 squares.

For 2.29, I'm confused where the 9 is coming from, as there are only 6 classes in a day. You can calculate the answer as$$\binom{6}{4}\cdot 4!\cdot 8\cdot 7$$where you choose 4 time periods for the math classes, then order the math classes, and then pick the 2 additional classes.