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January 21st, 2015, 05:02 PM   #1
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Permutation and Combinations Problem

I am trying to do an assignment for my current math class, and I have been stuck on it for hours.
Here is the question:

the 6 digits 0 through 5 can be arranged in a 2X3 table (a table with 2 rows and 3 columns). How many ways are there to do this if for each column, the entry in row 1 must be less than the entry in row 2?

the answer is 90, but I can't for the life of me figure out how.
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January 21st, 2015, 05:35 PM   #2
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Let's start by picking three numbers for the bottom row: $\{b_1, b_2, 5\}$ in ascending order. We'll then arrange the remaining three numbers above them so that we have three columns. Finally, we'll count the possible arrangements of the columns.

If $b_2 \ne 4$ we must have $b_2 = 3$ (the four must go above the 5 and then there is no number that can go below the 3). So there are only two options for $b_2$.

If $b_2 = 4$ then $b_1$ can be any of 1, 2 or 3.
If $b_1 = 3$ we can assign 0, 1 and 2 in any order to $a_1$, $a_2$ and $a_3$. There are 6 ways to order them.
If $b_1 = 2$, the 3 must go above either the 4 or the 5 (two choices) and the 0 and the 1 can be placed anywhere in the other two places (two choices). That's 4 ways to do it.
If $b_1 = 1$, the 0 must go above the 1 and then the other two can go in either of the remaining places. That's 2 ways.
That gives a total of 12 ways to have 4 and 5 on the bottom.

If $b_2 = 3$ then $b_1$ can be either 1 or 2. That's two choices.
If $b_1 = 2$ we must have the 4 above the 5 and the 0 and 1 can then be placed anywhere. So that is 2 ways to do it.
If $b_1 = 1$, the 0 must go above it and the 4 must go above the 5 so the 2 goes above the 3. That's 1 more way to do it.
That's 3 ways to have 5 but not 4 on the bottom.

Now we have 12+3=15 ways to create three columns. All we have to do is to arrange the columns. There are 6 ways to arrange the columns, so that's 6*15 = 90 ways to fill the grid as required.
Of those six ways,
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January 21st, 2015, 07:26 PM   #3
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There are 5 choices for the top row: {0, 1, 2}, {0, 1, 3}, {0, 1, 4}, {0, 2, 3} and {0, 2, 4}.
Note that these rows can be arranged in 3 nPr 3 = 6 ways.

For the table using the first top row, the numbers in the rows can be arranged in any
order so we have 6 * 6 = 36 possible arrangements.

For the table using the second top row, there are two places to place the 2 in the
bottom row and two places to place the 4 and the 5 (for each positioning of the 2),
so we have 6 * 2 * 2 = 24 possible arrangements.

For the table using the third top row, there are two ways to arrange the bottom row
({2, 3, 5} or {3, 2, 5}, for example), so we have 6 * 2 = 12 possible arrangements.

For the table using the fourth top row, there are two ways to arrange the bottom row
so we have 6 * 2 = 12 possible arrangements.

Finally, for the fifth top row, the bottom row can be arranged in only one way, so we
have 6 possible arrangements.

36 + 24 + 12 + 12 + 6 = 90.
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January 21st, 2015, 07:27 PM   #4
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Hello, jdesireg!

Quote:
The six digits 0 through 5 are arranged in a 2x3 table.
How many ways are there to do this if for each column,
the entry in row 1 must be less than the entry in row 2?

The answer is 90.

Suppose the first column is ${0\choose1}:\;\;\boxed{\begin{array}{ccc}0&*&* \\ 1&*&* \end{array}}$

The other four digits {2, 3, 4, 5} can be placed in three ways:

$\quad \boxed{\begin{array}{ccc}0&2&4 \\ 1&3&5 \end{array}}\qquad \boxed{\begin{array}{ccc}0&2&3 \\ 1&4&5 \end{array}} \qquad \boxed{\begin{array}{ccc}0&2&3\\ 1&5&4\end{array}}$


There are five choices for the first column: $\:{0\choose1},\;{0\choose2},\;{0\choose3},\;{0 \choose4},\;{0\choose5} $

The other two columns can be filled in three ways.

Then the three columns can be permutated in $3!=6$ ways.


Answer: $\:5\cdot3\cdot6 \:=\:90$ ways.

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January 22nd, 2015, 03:04 AM   #5
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The numbers in each of the columns can be arranged in 2 ways. So in three columns in 2^3 = 8 ways. The 6 numbers can be placed in 6! = 720 ways, of which 1/8 of them are as desired. So (720/8 = 90) ways.
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