Math Help - Permutations/combinations

1. Permutations/combinations

Question 1: Two dinner tables each hold eight people. In how many ways can a group of eight men and eight women divide themselsves so that there are equal number of men and women at each table?

Question 2: How many even numbers greater than 60, 000 can be formed with the digits {4, 5, 6, 7, 8} without repetitions?

Question 3: In how many ways is it possible to select seven letters including at least one I fom the word ILLUMINATI?

2. Originally Posted by yobacul
Question 1: Two dinner tables each hold eight people. In how many ways can a group of eight men and eight women divide themselves so that there are equal number of men and women at each table?
It is quite clear from the wording that there must be four of each sex at a table.
Here is a way to model this. Say that there is a lady A and a man X.
There are $\binom{7}{3}=35$ different grouping of four for A to be in.
That is, there are thirty-five pairing of four for the ladies.
That same logic holds for groups of four men containing X.
Now pair each female group containing A with a male group containing X.
Then pair each female group containing A with a male group not containing X.
How many such pairing do you get?
Is that the answer?

3. Hello, yobacul!

(2) How many even numbers greater than 60,000 can be formed
with the digits {4, 5, 6, 7, 8} without repetitions?

The first (leftmost) digit must be 6, 7, or 8.

Suppose the first digit is 7: . 7 _ _ _ _

The last digit must be even: {4, 6, 8} . . . 3 choices.

The remaining 3 digits can be placed in the middle 3 spaces: . $3!$ ways.

Hence, there are: . $3(3!) \:=\:18$ even numbers that begin with 7.

Suppose the first digit is 6 or 8: . $\begin{Bmatrix} _6\\^8\end{Bmatrix}\;\_\;\_\;\_\;\_$ . 2 choices.

The last digit must be even . . . 2 choices.

The remaining 3 digits can be placed in the middle 3 spaces: . $3!$ ways.

Hence, there are: . $2\cdot2\cdot3! \:=\:24$ even numbers that begin with 6 or 8.

Therefore, there are: . $18 + 24 \:=\:42$ even numbers greater than 60,000.