Here is how mr. K solved number 11.

11* A man and his wife invite 4 other couples to dinner. After the host and hostess have been seated at the ends of the table, in how many ways can the guests be seated so that no man sits beside another man or beside his own wife? (hint: make a diagram of the chairs and table; first find the number of ways of seating the ladies a, b, c, d; for a typical arrangement of the ladies, count the ways in which their husbands, A, B, C, D, can be assigned to the vacant chairs so that A and a B and b, ect. are not adjacent.)

If we had to seat all the couples at the tables it would be easiest to seat the men first. there are 4! ways to seat the men but if we chose one way for now these are the different seating arrangements we would have for the women.

After we seat the men there are 5 different ways to seat the women. mr. K mentioned this answer was a sexist answer because that is only the arrangements for the women so the final answer becomes (4!5)2 = 240.

During the afternoon this is what we put into our dictionary.

Solving Pathway Problems

This is Pascal's Triangle

Pascal's triangle turns up in many areas of math and science. It can be used to solve pathway problems as illustrated in these two examples:

Example 1:

There are 70 ways. The diagonals contain the respective entries from Pascal's triangle.

Example 2:

How many different ways can we spell the name Yves?

Solution: Using Pascal's triangle we find the sum of the entries in the bottom row 1 + 3 + 3 + 1 = 8 ways

Method 2

Solving Pathway Problems as Permutations of Non-distinguishable objects:

Find the number of ways to get from A to B using the shortest route possible:

Solution: each route goes right (R) 4 times and down (D) 3 times. Therefore each different arrangement of letters R R R R D D D counts as a different route.

This problem is the same as find the number of "different words" that can be spelled using the letters.

R R R R D D D

# of letters: 7 # of R's: 4 # of D's: 3

7! / 4! 3! = 7 * 6 * 5 * 4 * 3 * 2 * 1 / 3 * 2 * 1 * 4 * 3 * 2 * 1

=7 * 5 / 1

= 35

That is all we wrote in our dictionary for friday and im really sorry this took me a long time to post. hope this is helpful for the people who missed out on friday. Monday's scribe is Muuxi.

Wow! Great graphics! They're so clear and neatly written. Y'know, you should write a textbook. ;-)

BTW, the sexist answer was the one where the host sat at the head of the table and the hostess sat at the foot. If they can switch (the non-sexist solution) then we multiply by 2! because the hostess can choose from two seats -- ladies first ;-) -- followed by the host who can only choose one.

Posted by Mr. Kuropatwa | 3/05/2006 11:07 PM

## Post a Comment