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Ateacher takes 3 children from her class to the zoo at a time as often as she can, but she does not take the same three children to the zoo more than once. She finds that she goes to the zoo 84 times more than a particular child goes to the zoo. The number of children in her class is
- a)12
- b)10
- c)60
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
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Ateacher takes 3 children from her class to the zoo at a time as often...
Let the number of children in the class be n.
a particular child goes to zoo for n−1C2 times (from the remaining n−1 children , we select 2 children
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Ateacher takes 3 children from her class to the zoo at a time as often...
Given:
- A teacher takes 3 children from her class to the zoo at a time as often as she can, but she does not take the same three children to the zoo more than once.
- She finds that she goes to the zoo 84 times more than a particular child goes to the zoo.
To find: The number of children in her class
Solution:
Let's assume the number of children in her class is "n".
- Total number of ways to select 3 children out of "n" is given by nC3 = n(n-1)(n-2)/6
- The teacher goes to the zoo with these 3 children nC3 times.
- Since the teacher does not take the same three children to the zoo more than once, the number of times she goes to the zoo is limited by the child who goes to the zoo the least number of times.
- Let's assume this child goes to the zoo "x" times. Then the teacher goes to the zoo "x+84" times.
- So we can write the equation as: nC3 = x + 84
- We can also write the equation for the number of times the child goes to the zoo as: n-3C0 + n-3C1 + n-3C2 + ... + n-3Cn-3 = x
- Here, we are selecting 0 or 1 or 2 or ... or n-3 children from the remaining n-3 children to accompany the particular child to the zoo.
- Simplifying the second equation, we get: 2n-6 = 6x/n - x
- Substituting the value of x from the first equation, we get: 2n-6 = 6(nC3 - 84)/n - (nC3 - 84)
- Simplifying further, we get: (n-5)(n-6) = 504
- The only values of n that satisfy this equation are 10 and 11.
- However, if we take n=11, then nC3 would be 165 which cannot be expressed as x+84 for any value of x. Therefore, the only possible value of n is 10.
Answer: The number of children in her class is 10.
- A teacher takes 3 children from her class to the zoo at a time as often as she can, but she does not take the same three children to the zoo more than once.
- She finds that she goes to the zoo 84 times more than a particular child goes to the zoo.
To find: The number of children in her class
Solution:
Let's assume the number of children in her class is "n".
- Total number of ways to select 3 children out of "n" is given by nC3 = n(n-1)(n-2)/6
- The teacher goes to the zoo with these 3 children nC3 times.
- Since the teacher does not take the same three children to the zoo more than once, the number of times she goes to the zoo is limited by the child who goes to the zoo the least number of times.
- Let's assume this child goes to the zoo "x" times. Then the teacher goes to the zoo "x+84" times.
- So we can write the equation as: nC3 = x + 84
- We can also write the equation for the number of times the child goes to the zoo as: n-3C0 + n-3C1 + n-3C2 + ... + n-3Cn-3 = x
- Here, we are selecting 0 or 1 or 2 or ... or n-3 children from the remaining n-3 children to accompany the particular child to the zoo.
- Simplifying the second equation, we get: 2n-6 = 6x/n - x
- Substituting the value of x from the first equation, we get: 2n-6 = 6(nC3 - 84)/n - (nC3 - 84)
- Simplifying further, we get: (n-5)(n-6) = 504
- The only values of n that satisfy this equation are 10 and 11.
- However, if we take n=11, then nC3 would be 165 which cannot be expressed as x+84 for any value of x. Therefore, the only possible value of n is 10.
Answer: The number of children in her class is 10.
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Ateacher takes 3 children from her class to the zoo at a time as often as she can, but she does not take the same three children to the zoo more than once. She finds that she goes to the zoo 84 times more than a particular child goes to the zoo. The number of children in her class isa)12b)10c)60d)None of theseCorrect answer is option 'B'. Can you explain this answer?
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