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A certain list, L, contains a total of n numbers, not necessarily dist

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A certain list, L, contains a total of n numbers, not necessarily dist  [#permalink]

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New post 26 Apr 2019, 03:31
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A certain list, L, contains a total of n numbers, not necessarily distinct, that are arranged in increasing order. If \(L_1\) is the list consisting of the first \(n_1\) numbers in L and \(L_2\) is the list consisting of the last \(n_2\) numbers in L, is 17 a mode for L ?


(1) 17 is a mode for \(L_1\) and 17 is a mode for \(L_2\).

(2) \(n_1 + n_2 = n\)


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Re: A certain list, L, contains a total of n numbers, not necessarily dist  [#permalink]

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New post 26 Apr 2019, 04:29
A certain list, L, contains a total of n numbers, not necessarily distinct, that are arranged in increasing order. If \(L_1\) is the list consisting of the first \(n_1\) numbers in L and \(L_2\) is the list consisting of the last \(n_2\) numbers in L, is 17 a mode for L ?

Say the list is a, b, c, c, d, e, f, h, h, h, k, l..
\(L_1\) could be just a, and \(L_2\) could be h,h,k,l OR
\(L_1\) could be a,b,c,c,d and \(L_2\) could be c,c,d..h,h,k,l

(1) 17 is a mode for \(L_1\) and 17 is a mode for \(L_2\).
Since the list is in increasing order and mode is 17 in both cases, the list L1 and L2 either have all numbers within themselves OR only some other 17 are left.
For example 1,3,5,17,17,17,17,17,20,25....L1 could be 1,3,5,17,17 and L2 could be 3,5,17,17,17,17,17,20,25
OR L1 could be 1,3,5,17,17 and L2 could be 17,17,20,25
So the mode of L will be 17 in each case.
Suff

(2) \(n_1 + n_2 = n\)
We do not know the numbers.
Insuff

A
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A certain list, L, contains a total of n numbers, not necessarily dist  [#permalink]

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New post Updated on: 29 Apr 2019, 08:18
1
The Logical approach to this question will use the definition of 'mode' - the most frequent value of a set of data.
In statement (1),if the number 17 appears more than all other numbers in both of the list, it must also appear more than any other number in the combined list as well. It is like saying that if a > x and b > y, then a + b > x + y: if 17 appears more than the second frequent number in both lists, it will appear more than the second greatest number in the combined list as well. This statement is enough on its own.
Statement (2) does not tell us which numbers are in the set, so is insufficient.
The correct answer is (A).

Posted from my mobile device

EDIT: *kudos to Shukhrat below for pointing out an error in my explanation of stmt (2) (which did not change the final answer)
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Originally posted by DavidTutorexamPAL on 27 Apr 2019, 14:13.
Last edited by DavidTutorexamPAL on 29 Apr 2019, 08:18, edited 2 times in total.
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A certain list, L, contains a total of n numbers, not necessarily dist  [#permalink]

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New post Updated on: 06 May 2019, 00:18
Bunuel wrote:
A certain list, L, contains a total of n numbers, not necessarily distinct, that are arranged in increasing order. If \(L_1\) is the list consisting of the first \(n_1\) numbers in L and \(L_2\) is the list consisting of the last \(n_2\) numbers in L, is 17 a mode for L ?


(1) 17 is a mode for \(L_1\) and \(17\) is a mode for \(L_2\).

(2) \(n_1 + n_2 = n\)


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Hola amigos :cool:

I disagree with David from examPAL.

Mode - is the number that occurs most often. If no number in the list is repeated, then there is no mode for the list.

1. \(17\) is a mode for \(L_1\) and \(17\) is a mode for \(L_2\).
Most important thing to note is the numbers are in increasing order. That means that there are only \(2\) possible cases:
Case 1: \(L_1\) and \(L_2\) togther include all the numbers \(L\) has. E.g. \(L\) = {1, 1, 3, 17, 17, 17, 19, 21,}, \(L_1\) = {1, 1, 3 , 17, 17, 17, 19} and \(L_2\) = {3, 17, 17, 17, 19, 21}
Case 2: The numbers of \(L\) not included in \(L_1\) and \(L_2\) are all \(17\). E.g. \(L\) = {1, 1, 3, 17, 17, 17, 17, 17, 17, 17, 19, 21,}, \(L_1\) = {1, 1, 3, 17, 17, 17} and \(L_2\) = {17, 17, 19, 21} Those two reds are leftover.
In both cases \(17\) is the most repeated number and there is no other possible case, thus the mode of \(L\) is \(17\).
Sufficient

2. \(n_1+n_2=n\)
The given information is not repeated in other words here. We already have seen that some \(17\) can be left. Though no info regarding numbers here.
Insufficient

The answer is A
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Originally posted by ShukhratJon on 29 Apr 2019, 06:14.
Last edited by ShukhratJon on 06 May 2019, 00:18, edited 1 time in total.
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Re: A certain list, L, contains a total of n numbers, not necessarily dist  [#permalink]

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New post 05 May 2019, 15:34
Bunuel wrote:
A certain list, L, contains a total of n numbers, not necessarily distinct, that are arranged in increasing order. If \(L_1\) is the list consisting of the first \(n_1\) numbers in L and \(L_2\) is the list consisting of the last \(n_2\) numbers in L, is 17 a mode for L ?

(1) 17 is a mode for \(L_1\) and 17 is a mode for \(L_2\).

(2) \(n_1+n_2=n\)

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Three cases are possible about the mode of a list of numbers. If the numbers are all distinct, then the list has no mode. If one number's frequency is greater than the frequencies of all other numbers, then the list has one mode. However, it's also possible that a list of not all distinct numbers has multiple modes if two or more numbers have the greatest frequency simultaneously.

We know that \(L\) is a list of numbers, not necessarily distinct, that are arranged in increasing order. The original question: Is 17 a mode for \(L\)?

1) We can infer that in the list no number under 17 has greater frequency than that of 17. We can also infer that in the list no number above 17 has greater frequency than that of 17. Using that no number in the list has greater frequency than that of 17 and that 17 has a frequency of at least 2, we can answer the original question with a definite Yes. \(\implies\) Sufficient

2) No information is given about the actual numbers. If all numbers in the list are 17, then the answer to the original question is Yes. However, if 17 is not even in the list, then the answer to the original question is No. Thus, we can't get a definite answer to the original question. \(\implies\) Insufficient

Answer: A
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Re: A certain list, L, contains a total of n numbers, not necessarily dist  [#permalink]

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New post 10 May 2019, 16:16
Hi All,

We're told that List L contains a total of N numbers, NOT necessarily DISTINCT, that are arranged in INCREASING order and L1 is the list consisting of the first N1 numbers in L and L2 is the list consisting of the last N2 numbers in L. We're asked if 17 is a MODE for L. It's worth noting that the two sub-lists (L1 and L2 could potentially "overlap", meaning that some numbers of List L appear in BOTH groups). Every number in List L is accounted for though (again though, it's possible that some values appear in both sub-lists).

To start, it's worth noting that a "mode" is a number that shows up the most often in a group of numbers (for example, the group 1, 2, 2, 3, 4 has a mode of 2). There can actually be more than one mode in a group though (for example, the group 1, 1, 2, 2, 3, 4 has two modes: 1 and 2).

(1) 17 is a mode for L1 and 17 is a mode for L2.

Fact1 tells us that 17 is a mode for both sub-lists. Note that the prompt did tell us that the list is arranged in INCREASING order...
IF....
-there is NO 'overlap', then some of the 17s appear in L1 and some appear in L2. Totaling all of those individual 17s would make the mode even larger than it is in the two individual sub-lists, meaning that no other value could appear more often in List L.
-there IS an 'overlap' - meaning one (or more) of the 17s appears in BOTH sub-lists, then that total number of 17s will still be a mode. There might be other values that are also modes, but those values could not show up more often than the 17s do, so none of them could "out-number" the 17s.
Fact 1 is SUFFICIENT

(2) N1 + N2 = N

Fact 2 tells us that the two sub-lists have no terms in common (meaning that there is no 'overlap'), but it does not tell us anything about the individual terms or what the mode(s) might be.
Fact 2 is INSUFFICIENT

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Re: A certain list, L, contains a total of n numbers, not necessarily dist  [#permalink]

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New post 15 May 2019, 00:14

Solution



Steps 1 & 2: Understand Question and Draw Inferences
In this question, we are given
    • A certain list, L, contains a total of n numbers, not necessarily distinct, that are arranged in increasing order.
    • L1 is the list consisting of the first n1 numbers in L.
    • L2 is the list consisting of the last n2 numbers in L.

We need to determine
    • Whether 17 is a mode for L or not.

Step 3: Analyse Statement 1
As per the information given in statement 1, 17 is a mode for L1 and 17 is a mode for L2.
Therefore, we can infer that
    • 17 must occur in L1, either same or a greater number of times as any other number in L1.
    • 17 must occur in L1, either same or a greater number of times as any other number in L2.

As all elements in L are in ascending order, we can also conclude that
    • Each number between last occurrence of 17 in L1 and the first occurrence of 17 in L2 must be equal to 17 only.
    • Therefore, 17 occurs either same or greater number of times as any other number in L.
      o Thus, 17 is a mode for L.

Hence, statement 1 is sufficient to answer the question.

Step 4: Analyse Statement 2
As per the information given in statement 2, n1 + n2 = n.
    • However, from this statement, we cannot conclude anything about the mode of L1, L2, or L.

Hence, statement 2 is not sufficient to answer the question.

Step 5: Combine Both Statements Together (If Needed)
Since we can determine the answer from statement 1 individually, this step is not required.

Hence, the correct answer choice is option A.

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Re: A certain list, L, contains a total of n numbers, not necessarily dist   [#permalink] 15 May 2019, 00:14
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