Answer to Question #283056 in Combinatorics | Number Theory for Tylor

Question

1. How many 4 digits numbers can we create from 3,4,5,6,7,0,9 if;
2. We need all numbers to be even numbers        [2]
3. We need all of them to be odd numbers        [2]
4. We need all of them to be divisible by 5        [3]
5. In how many different ways can we choose a committee that is comprised of two men and one woman from a group of 5 men and 4 women if;
6. We need a chosen committee member is selected only once (i.e. if a person occupies only one position in the committee)                [2]
7. We need to choose a chairperson, a treasurer and secretary [3]

In how many ways can we arrange the letters of the word EXCELLENT to form different 9 letter words, with or without meaning   

Answer 1

1.

2. The first place can be occupied by any of the six numbers. The second place can be occupied by any of the seven numbers. The third place can be occupied by any of the seven numbers. The fourth place can be occupied by any of the three numbers.

Therefore total number of 4 digit numbers in this case is

$$6(7)(7)(3)=882$$

3. The first place can be occupied by any of the six numbers. The second place can be occupied by any of the seven numbers. The third place can be occupied by any of the seven numbers. The fourth place can be occupied by any of the four numbers.

Therefore total number of 4 digit numbers in this case is

$$6(7)(7)(4)=1176$$

4. The first place can be occupied by any of the six numbers. The second place can be occupied by any of the seven numbers. The third place can be occupied by any of the seven numbers. The fourth place can be occupied by any of the two numbers.

Therefore total number of 4 digit numbers in this case is

$$6(7)(7)(2)=588$$

5.

If the order does not matter

$$\dbinom{5}{2}\dbinom{4}{1}=10(4)=40$$

6.

$$\dbinom{5}{2}\dbinom{4}{1}=10(4)=40$$

7. A chairperson is a men , a treasurer is a men and secretary is a women

$$5(4)(4)=80$$

A chairperson is a men , a treasurer is a women and secretary is a men

$$5(4)(4)=80$$

A chairperson is a women , a treasurer is a men and secretary is a men

$$4(5)(4)=80$$

The total number is

$$80+80+80=240$$

8. There are 3 E’s, 2 L’s, 1 X's, 1 C's, 1 N's and 1 T’s in this word, therefore, the number of ways of arranging the letters are:

$$\dfrac{9!}{3!2!1!1!1!1!}=30240$$