chetan2u 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 ?
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
Hi
chetan2u , thanks for the explanation
I thought (1) is insufficient, because there is no indication that n_1 and n_2 is adjacent. In other words, I concerned that there is a gap between n_1 and n_2.
If the gap exists, the don't know how big it is or what consists the gap.
Please help me what I missed
Thank you in advance!!
You bring up a good point: what if there is a group of numbers BETWEEN group L1 and group L2? At this point, what additional work did you do when you worked through Fact 1...?
If there is a set of numbers that is not in either L1 or L2, how would it be possible for 17 to be the MODE for BOTH of those groups (since there are numbers 'in between' the two groups, how could 17 be in BOTH groups?)? Remember that the numbers are arranged in order from least to greatest and there could be duplicates.....
There is only one way for that to occur: if ALL of the numbers that are NOT in L1 nor in L2 are all 17s. In this situation, 17 would still be the mode for the entire list and the answer to the question would still be 'YES.'