by using inclusion exclusion principle and derangements find the solution?

Question

. A survey was done on 1400 workers of a factory and shows that  675 are males  682 range from 19 to 20 years old  684 are in sports/recreational activities  195 are males and are from 19 to 20 years old  467 are males in sports/recreational activities  318 are in sports/recreational activities and range in age from 19 to 20 years  165 are males in sports/recreational activities and range in age from 19 to 20 years (a) In this group of 1400 factory workers, how many were females who are not aged 19 or 20, and who are not in sports/recreational activities? (b) In this group of 1400 factory workers, how many males not involved in sports/recreational activities range in age from 19 to 20 years? (c) In this group of 1400 factory workers, how many females not involved in sports/recreational activities were questioned? [Hint: Draw a Venn diagram with three sets each representing “males’, “age 19 to 20” years old and in “sports/recreational activities”. 2. Suppose you want to permute the first five positive integers. How many ways are there if (a) you require the third integer to be in its natural position but the others are not. (b) you require the third integer to be in its natural position and exactly one other integer in its natural position as well, but the others are not. 

Answer 1

QUESTION 1

First the Venn diagram:

 

KEY

A Females/outside 19-20 age group/not involved in sports or recreation activities

B Males only

C 19-20 year old females only 

D Involved in Sports/Recreation (S/R) only

E Males involved in S/R, but not 19-20 years old

F 19-20 year olds involved in S/R

G 19-20 year old males not involved in S/R

H 19-20 year old males involved in S/R

A+B+C+D+E+F+G+H=1400 (all employees)

B+E+G+H=675 (males)

C+F+G+H=682 (19-20 year olds)

D+E+F+H=684 (involved in S/R)

G+H=195 (19-20 year old males)

E+H=467 (males in S/R)

F+H=318 (in S/R)

H=165 (19-20 year old males in S/R)

(a) Requires us to find A.

Given H=165 we can find F=153, E=302, G=30.

Given E=302, F+H=318, D=64.

Given F=153, G+H=195, C=334.

Given E=302, G+H=195, B=178.

Now we can find A=1400-(178+334+64+302+153+195)=174.

(b) To find G; G=30.

(c) To find C; C=334.

QUESTION 2

(A) First 5 integers 1 2 3 4 5.

There are 24 ways of arranging 1 2 4 5 including the numbers being in their natural position, so we need to exclude permutations where that would happen.

24 permutations:

1245 1254 1425 1452 1524 1542 

2145 2154 2415 2451 2514 2541

4125 4152 4215 4251 4512 4521

5124 5142 5214 5241 5412 5421 

9 remain not crossed out, so there are 9 permutations with only 3 in its natural position.

(b) There are 8:

41325, 51342, 24315, 52314, 14352, 42351, 15324, 25341