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So, 20 arrangements where AI initiatives are adjacent.
Step 3: Subtract to get valid (non-adjacent) sequences
Therefore, the number of valid sequences where no two AI security initiatives are adjacent is:
Question: An angel investor is considering a portfolio of 8 biotech firms: 4 developing mRNA vaccines, 3 working on CRISPR therapies, and 1 focused on regenerative medicine. If the investor selects 4 firms at random, what is the probability that at least one firm from each research area is included?
Total number of possible selections:
Since there are 3 areas and we choose 4 firms, the only way to include all three areas is to have one of the following distributions:
$ (2,1,1) $: two from one area, one each from the other two
So only valid composition is permutations of $ (2,1,1) $. We compute for each case.
So the probability that at least one firm from each area is included is:
Question: A science fair judge must assign 6 identical ribbons (gold, silver, and bronze — with 2 gold, 2 silver, and 2 bronze) to 6 winning projects, each receiving exactly one ribbon. If the ribbons of the same type are indistinguishable, how many distinct ways can the ribbons be distributed?