Probability

evene · 2016-03-04 12:14:52

1.) In how many ways can you spell the word COOL in the grid below? You can start on any letter, then on each step you can step one letter in any direction (up, down, left, right, or diagonal).

C C C C C
L O O O L
L O O O L
L O O O L
C C C C C

2.) I have 7 different math books and 4 different history books. In how many ways can I line the 11 books up on a shelf if a history book must be in the middle and each end must be a math book?

Grantingriver · 2016-03-04 13:28:27

1) Each of the the two Cs in the bottom row left to the C in the middle has 3 ways to spell "COOL" and the same for the two right Cs while the C in the middle has 5 ways to do that hence the total is 17 and since we have the same situation in the top row, therefore the grand total is "34" ways.

2) Since the first and last books must be math and the middle book is history, then we must fix and subtract these three books because all the permutations will have the same form in the middle and ends hence we will be left with 8 books (5 math and 3 history) therefore the total ways "W1":
W1= 8!=8×7×6×5×4×3×2×1=40320 ways.
On the other hand, since we have 7 "different" math books and 4 "different history books we can choose the fixed ends and middle books respectively in the following ways:
W2= 7×6=42 and W3=4 hence the grand total ways "W" is:
W=W1×W2×W3=40320×42×4=6773760 ways.

Last edited by Grantingriver (2016-03-04 13:57:28)

Nehushtan · 2016-03-04 23:06:58

Grantingriver wrote:

1) Each of the the two Cs in the bottom row left to the C in the middle has 3 ways to spell "COOL" and the same for the two right Cs while the C in the middle has 5 ways to do that hence the total is 17 and since we have the same situation in the top row, therefore the grand total is "34" ways.

Totally wrong.

Label the letters like this

[list=*]
[*]

[/*]
[/list]

Let's start with C1. There are 3 ways.

[list=1]
[*]

[/*]
[*][/*]
[*][/*]
[/list]

By symmetry there are also 3 ways starting with C5, C6 and C10. This gives 12 ways so far.

Now let's start with C2.

[list=1]
[*]

[/*]
[*][/*]
[*][/*]
[*][/*]
[*][/*]
[*][/*]
[*][/*]
[*][/*]
[*][/*]
[*][/*]
[*][/*]
[*][/*]
[*][/*]
[/list]

By symmetry, there are also 13 ways starting with C4, C7 and C9. So there are 52 ways, giving 64 ways so far.

Now let's start with C3. If the second letter is O1 or O2, we get 13 ways as if we start with C2. If we go C3O3, we get 3 ways as if we start with C5. This gives us 16 ways. By symmetry there are also 16 ways starting with C8. So there are 32 ways starting with the middle top or bottom C.

Hence there are exactly 96 ways to spell COOL.

Last edited by Nehushtan (2016-03-04 23:21:21)

Grantingriver · 2016-03-05 01:42:52

Sorry, but it is not "totally wrong" the first parts are correct in both arguments:

Nehushtan wrote:

Let's start with C1. There are 3 ways...By symmetry there are also 3 ways starting with C5, C6 and C10. This gives 12 ways so far.

And:

Grantingriver wrote:

1) Each of the the two Cs in the bottom row left to the C in the middle has 3 ways to spell "COOL" and the same for the two right Cs

So the corners have the same number of ways in both arguments, also you have devised the same technique.

Grantingriver wrote:

since we have the same situation in the top row

By saying "by symmery". So the mistake just in "counting" which is a trivial one.

evene · 2016-03-05 02:25:43

Ok, thanks guys!

Math Is Fun Forum

#1 2016-03-04 12:14:52

Probability

#2 2016-03-04 13:28:27

Re: Probability

#3 2016-03-04 23:06:58

Re: Probability

#4 2016-03-05 01:42:52

Re: Probability

#5 2016-03-05 02:25:43

Re: Probability

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