Chris visited the local pizza shack to purchase one pizza for each member of his scout troop. on arrival, he realised that he had mislaid the list he had made of the pizzas that had been requested. he knew that there were 32 people altogether and that all 3 vegetarians in the troop wanted mushrooms only. he also remembered that the same number of people wanted just the chicken topping as wanted a plain pizza. alex, a young leader, was able to by recalling that: 12 people requested toppings that included chicken, 10 requested toppings that included pepperoni, 9 requested toppings that included mushroom, 1 person wanted all 3 toppings, 2 wanted chicken and pepperoni only, 2 wanted peperoni and mushroom only. what was the final order that chris placed at pizza shack?

From the information provided in the question the following assumptions can be made;
32 people altogether
1 requested all three toppings
3 vegetarians got mushroom only 
9 requested toppings with mushrooms included
12 requested toppings with chicken included
10 requested toppings that included pepperoni 
2 wanted chicken and pepperoni only 
2 wanted mushroom and pepperoni
x amount of people wanted chicken only
x amount of plain pizzas
all this information is depicted in the diagram attached

the amount of pepperoni only = 10 - 1 - 2 - 2
                                               = 5
the amount of mushroom only (y)
the number of pizza with both chicken and mushroom (z)
y = 9 - 1 - 2 -z
  = 6-z
amount of chicken toppings only 
x = 12 - 2 - 1 - z 
   = 9-z
total number of pizzas 
32 = 1 + 2 + 2 + 5 + y + x  + z +x
     = 10 + y + 2x +z
substiuting y = 6 - z 
   32 = 10 + 6 - z + 2x + z
   32= 16 + 2x
   2x = 16
    x = 8
then x = 9-z
        8 = 9-z
        z = 1
 and y = 6 - z
       y = 5

therefore the final answer was supposed to be 
8 plain pizzas
8 chicken only
5 pepperoni only
5 mushrooms only 
1 all toppings
2 chicken and pepperoni
2 mushroom and pepperoni 
1 chicken and mushroom

The answer of this question is 672 .

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