Children 82574

Twenty-nine children attend the club. Eleven said that they have a dog, 14 children have a cat, and 12 children have a hamster. Two children have all three animals. Seven children do not have any animals at home.
How many children have at least two of the listed animals? How many children does one of the animals have?

Correct answer:

x =  13
y =  9

Step-by-step explanation:

$$\begin{gathered} n=29 \\ {n}_0=7 \\ {n}_1=11 \\ {n}_2=14 \\ {n}_3=12 \\ {n}_{123}=2 \\ a=3 \\ b=2 \\ {n}_1=A+a+c+2 \\ {n}_2=b+c+2+C \\ {n}_3=a+b+2+B \\ A+B+C+a+b+c+2=n-n_0 \\ 11=A+3+c+2 \\ 14=2+c+2+C \\ 12=3+2+2+B \\ A+B+C+3+2+c+2=29-7 \\ A+c=6 \\ C+c=10 \\ B=5 \\ A+B+C+c=15 \\ Row4-Row1\to Row4 \\ A+c=6 \\ C+c=10 \\ B=5 \\ B+C=9 \\ Pivot:Row2\leftrightarrow Row3 \\ A+c=6 \\ B=5 \\ C+c=10 \\ B+C=9 \\ Row4-Row2\to Row4 \\ A+c=6 \\ B=5 \\ C+c=10 \\ C=4 \\ Row4-Row3\to Row4 \\ A+c=6 \\ B=5 \\ C+c=10 \\ -c=-6 \\ c=\frac{-6}{-1}=6 \\ C=10-c=10-6=4 \\ B=\frac{5}{1}=5 \\ A=6-c=6-6=0 \\ A=0 \\ B=5 \\ C=4 \\ c=6 \\ x=a+b+c+2=3+2+6+2=13 \end{gathered}$$

$y=n-n_0-x=29-7-13=9$

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