jeff goldbloom
,,,,,..
Periodic table of elements .
.
1183D
118 .
.
Z .
22.98976...
118
Related Articles
Trending Articles
Solution: We are distributing 6 distinguishable fish species into 4 indistinguishable zones, with each zone receiving at least one species. This is equivalent to finding the number of ways to partition a set of 6 labeled elements into exactly 4 non-empty unlabeled subsets. This is given by the Stirling numbers of the second kind, denoted $ S(n, k) $. Here, $ n = 6 $, $ k = 4 $.
We compute $ S(6, 4) $. Using known values or recurrence:
with base cases $ S(n, n) = 1 $, $ S(n, 1) = 1 $, and $ S(n, k) = 0 $ for $ k > n $.
Question: A marine biologist records daily sea surface temperatures over a 7-day period, classifying each day as warm (W), moderate (M), or cold (C). How many distinct 7-day temperature sequences contain exactly two W days, at least one M day, and no two W days are adjacent?
Solution: We are to count the number of sequences of length 7 using symbols W, M, C such that:
The remaining 5 days are filled with M and C (but not all C)
Step 1: Choose positions for the 2 non-adjacent W days in 7 positions.
Indeed, this counts the number of ways to place 2 indistinct non-adjacent markers in 7 slots. So there are 15 valid position sets for W.
Subtract the one case where all 5 are C: $ 32 - 1 = 31 $ valid assignments.
Thus, total sequences: $ 15 imes 31 = 465 $.