Test by Bikram | Mock GATE | Test 1 | Question: 62

Question

Consider the set of numbers $N= 1,2,3,4,5,6,7,8$. Take every distinct two-element subset of $N$ and write down the number that is smaller. For eg, if you take the subset$\left ( 2,5 \right )$, you will write down $2$.

The sum of all the numbers that you write down is ___________.

Answer 1

The possible pairs are (1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8).

(2,3),(2,4),(2,5),(2,6),(2,7),(2,8)

(3,4),(3,5),(3,6),(3,7),(3,8)

(4,5),(4,6),(4,7),(4,8)

(5,6)(5,7)(5,8)

(6,7),(6,8)

(7,8)

Therefore total sum =(1*7)+(2*6)+(3*5)+(4*4)+(5*3)+(6*2)+(7*1)

                               =84

Comments

(2,3) AND (3,2) ARE THEY SAME???

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Yes,because from the subset you have to choose the minimum value only.

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But, in the exam how can we differentiate , that we need to take (2,3) only and not (3,2)?

Is there any Hint itself in the Question?

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@Jason GATE You can take any one of them, it’s up to your understanding but you have to make sure that you don’t count it twice. In this answer, they used the basic approach of choosing a number and making it’s set with all the numbers greater than them. This is the simplest and the most basic method to avoid counting any set twice. Just invert all the subset element positions in this pattern and you will get the correct pattern with (3,2) ;)
 

Answer 2

Order

The definition of set states that a "set is an unordered collection of elements. What does this mean about sets C and D, below?

C={10,27,4}D={4,10,27}

Sets C and D are equal because there elements are the same. The only difference is the order, which doesn't matter.

So   84 is correct.

Comments

Got it !!! Thanks