A problem with my reasoning in a problem about combinations

Solution 1:

Answer:

Your method has two errors. First you are selecting one boy from 6 and the other four from the whole set of 10 (girls and boys). Further you are only selecting one boy when you should select all combination of 1, 2, 3, 4, 5 boys. Instead circumventing this route, you select five people from 11 people and subtract the term with zero boys Thus

it is either: ${11\choose5} - {5\choose5}$ or

it is $${6\choose1}{5\choose4}+{6\choose2}{5\choose3}+{6\choose3}{5\choose2}+{6\choose4}{5\choose1}+{6\choose5}{5\choose0}$$

Thanks

Satish

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