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A mining engineer is evaluating the environmental impact of mining operations and needs to assess 7 different methods for reducing emissions. They plan to test 4 methods simultaneously. In how many different ways can the engineer choose 4 methods from the 7?
To find the number of ways to choose 4 methods from 7, we use the combination formula:
Hence, there are 35 different ways to choose 4 methods from 7. Therefore, the final answer is:
A linguist is analyzing the evolution of language and considers 6 distinct phonemes. If the linguist wants to study permutations of 4 phonemes at a time, how many different permutations can be arranged from the 6 phonemes?
To calculate the number of permutations of 6 phonemes taken 4 at a time, we use the permutation formula:
Thus, there are 360 different permutations of 4 phonemes that can be chosen from the 6. Therefore, the final answer is:
If a mining engineer simulates the random selection of 2 out of 6 potential sustainable mining methods, what is the probability that a specific pair, say Methods A and B, is chosen?
First, we find the total number of ways to select 2 methods out of 6, using the combination formula:
There is only 1 favorable outcome where Methods A and B are chosen. Therefore, the probability of selecting Methods A and B is:
Thus, the probability that Methods A and B are chosen is: