QR combination question

[IdahoDoc]

For some reason I am really bad at these questions so thanks for the help.


Martha invited 4 friends to go with her to the movies. There are 120 different ways in which they can sit together in a row. In how many of those ways is martha sitting in the middle?


The answer is....




24 but i am not sure how to get it.


my next question is

how many 3-digit positive integers are odd and do not contain the digit "5"?



answer is...



288

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[TKDentist]

Mathematically, as best I can:

There are 5 people. Martha has to be in the middle (XXMXX) so what it is saying is that the other four people could be anywhere on the outsides.

- - M - -
4*3*1*2*1 = 24. As you place the first person, you can pick from anyone, and as you get down the line, you have less people to place: 3,2,1 respectively. Martha's seat counts as "1" because only she can sit there.

essentially you could start anywhere, therefore, you can do it in any order (hence why it is multiplication, if that helps you see it):

4*3*1*2*1 = 24

3*4*1*2*1 = 24

3*2*1*1*4 = 24

etc...

NEXT ONE:

okay, so you have 3 digits your working with:

- - -

they can't contain 5, and since this is the first, you can't have zero either. so your left with 8 digits to work with for the first digit:

8 - -

the second CAN include zero, so there are 9 digits to work with for the second one:

8*9 -

the third digit has to be odd, so therefore there are only 4 choices, since 5 isn't allowed (1,3,7,9). so you have 4 digits to choose for the third digit:

8*9*4 = 288.

hope this helps.

 

[sfoksn]

you are good lol

 

[TKDentist]

you are good lol

thanks. I think of it mathematically and visually at the same time. it's weird. lol. but glad you think so.