Question

For a natural number \[n\] , Let \[T(n)\] denote the number of ways we can place n objects of weights \[1,2,3,....,n\] on a balance such that the sum of the weights in each pan is the same then \[T(100) > T(99)\] .
A. True
B. False

Answer 1

Hint: To check that the given statement is true or false. First understand the definition and meaning of the given function \[T(n)\] . Here we have to show that the number of all the possible causes for \[T(99)\] is less than the number of causes for \[T(100)\] .

Complete step by step solution:
Since, here \[T(n)\] is a function of \[n\] is equal to the number of ways of splitting \[n\] into 2 parts such that each part contains \[1,2,3,....,n\] denomination.
So \[T(99)\] can be like \[\left( {1,4} \right)\] and \[(5)\] , \[\left( {2,4} \right)\] and \[(6)\] , \[\left( {2,3} \right)\] and \[(5)\] and many more
Hence all the possible causes for are also favorable causes (lies causes) for \[T(100)\] because the set \[\left( {1,2,3,....,99} \right)\] is a subset of the set \[\left( {1,2,3,....,100} \right)\] and \[100\] is extra denomination for \[T(100)\] . So \[T(100)\] can be used to form like \[\left( {48,49,3} \right)\] and \[\left( {100} \right)\] and many more.
Hence, \[T(100) > T(99)\]
The given state is true.
So, the correct answer is “Option A”.

Note: Note that many times students understand the wrong meaning and definition of the given function. They get another answer and they think the given statement is wrong.
Hence clarity about which function is subset of