A computer network administrator needs to assign unique identification

Question

A computer network administrator needs to assign unique identification codes to each of the 16 network nodes. Each code is either a single digit or a two-digit sequence, where the two digits are distinct and must appear in strictly increasing order (for example, 3 followed by 7). What is the minimum number of different digits needed to generate at least 16 distinct codes?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Krunaal · Answer

Let's suppose we have  n  different digits, then each of the  n  digit, plus  \frac{n*(n-1)}{2} , since each of the different digits will get paired with one other digit only in one way (i.e. in ascending order).

So, applying  n + \frac{n*(n-1)}{2}  to the options, we find out that we need at least 6 different digits for 16 distinct codes.

Answer C.

scream7 · Answer

Theory :

If we have 5 digits, and we need to make a unique one digit code; there will be 5C1 ways = 5; (since they are in ascending order, only 1 way to arrange them)
If we have 5 digits, and we need to make a unique two digit code; there will be 5C2 ways = 10 x 1 arrangement
So total =15

Now, lets try from the options :
Since 5 digits total makes 15

Lets try 6 -->
1-digit : 6C1 = 6
2-digit : 6C2 = 15
Total = 21 which is enough for 16 network nodes (question).

advaithj · Answer

How can we take 5C2? It is mentioned in the question that the 2 digit sequence must be in the increasing order. Won't the 5C2 arrangment include other arrangements as well ? same goes for 6C2 ? Please explain ?

Krunaal · Answer

5C2, and 6C2 are selections, and not arrangements. You're just selecting 2 numbers out of 5, or 6. Let's suppose the 5 numbers are {1,2,3,4,5}, you can select two numbers in 10 ways {1,3}, {2,5} and so on and so forth. If you select {1,3} you cannot select {3,1} since that would be an arrangement.

If it were 5P2 or 6P2, or 5C2*2! that would give arrangements eg. {1,3} and {3,1} both would be included in that, that's not the case here.

A computer network administrator needs to assign unique identification

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A computer network administrator needs to assign unique identification

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