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# Counting

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You are given the 4 x 4 grid below.

(a) Find the number of ways of placing 8 counters in the squares (at most one counter per square), so that each row contains exactly two counters.

(b) Find the number of ways of placing 12 counters in the squares (at most one counter per square), so that each column contains exactly three counters.

Mar 13, 2023

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(a) To place 8 counters in the 4 x 4 grid so that each row contains exactly two counters, we can proceed as follows:

- Choose any two squares in the first row and place counters in them. There are (4 choose 2) = 6 ways to do this.
- Choose any two squares in the second row that do not already have counters and place counters in them. There are (2 choose 2) = 1 way to do this.
- Choose any two squares in the third row that do not already have counters and place counters in them. There are (2 choose 2) = 1 way to do this.
- Place the remaining two counters in the fourth row. There is only one way to do this.

Therefore, the total number of ways to place 8 counters in the 4 x 4 grid so that each row contains exactly two counters is 6 x 1 x 1 x 1 = 6.

(b) To place 12 counters in the 4 x 4 grid so that each column contains exactly three counters, we can proceed as follows:

- Choose any three squares in the first column and place counters in them. There are (4 choose 3) = 4 ways to do this.
- Choose any three squares in the second column that do not already have counters and place counters in them. There are (3 choose 3) = 1 way to do this.
- Choose any three squares in the third column that do not already have counters and place counters in them. There are (2 choose 3) = 0 ways to do this, since there are only two squares left in the third column.
- Place the remaining three counters in the fourth column. There are (3 choose 3) = 1 way to do this.

Therefore, the total number of ways to place 12 counters in the 4 x 4 grid so that each column contains exactly three counters is 4 x 1 x 0 x 1 = 0. There is no way to do this.

Mar 13, 2023