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Thus, the total number of distinct seating arrangements is:
A team of 3 historians and 5 engineers is tasked with organizing seating for a round conference table. If the 3 historians must sit consecutively, in how many ways can they be seated?
Consider the 3 historians as a single block. This reduces the problem to arranging 6 units around the table: the block of historians and the 5 engineers.
Thus, the total number of seating arrangements is:
In a virtual exhibit, a museum curator wants to display 4 paintings and 2 sculptures. If the paintings must always be grouped together, how many ways can the items be arranged in a line?
Consider the 4 paintings as a single super item. This transforms the problem into arranging 3 items: the super item and the 2 sculptures.
A research grant requires that a number of distinct positive integers each less than or equal to 50 be selected such that each is congruent to 3 modulo 7. What is the total number of such integers?
So $ k = 0, 1, 2, 3, 4, 5, 6 $, yielding 7 values.
In a carbon capture model, a nanotech matrix cycles through states labeled by integers from 1 to 100. The system activates only when the sum of three consecutive integers in this range is divisible by the smallest prime factor of 210. What is the smallest such prime factor, and how many such triples exist?
But $ 3n $ is even if and only if $ n $ is even.