A DOH study concluded that by following 7 simple health rules,a man\'s life can

Question

A DOH study concluded that by following 7 simple health rules,a man's life can be extended by 11 years on the average and awoman's life by 7 years. These 7 rules are: no smoking, regularexercise, use alcohol moderately, get 7 to 8 hours of sleep,maintain proper weight, eat breakfast, and do not eat betweenmeals. In how many ways can a person adopt 5 of these rules tofollow; a) if the person presently violates all 7 rules? b) if a person never drinks and always eats breakfast? A DOH study concluded that by following 7 simple health rules,a man's life can be extended by 11 years on the average and awoman's life by 7 years. These 7 rules are: no smoking, regularexercise, use alcohol moderately, get 7 to 8 hours of sleep,maintain proper weight, eat breakfast, and do not eat betweenmeals. In how many ways can a person adopt 5 of these rules tofollow; a) if the person presently violates all 7 rules? b) if a person never drinks and always eats breakfast?

Explanation / Answer

This is a question about COMBINATORICS.  We're givena "set" of size 7, and asked how many ways to choose FIVE elementsout of SEVEN. . For example, we could choose to adopt the rules on smoking,exercise, alcohol, sleep, and weight. Or we could choose toadopt the rules on smoking, exercise, alcohol, sleep, andbreakfast. . First, let me try to review some intuition that hopefully yourclass went over. Then I'll give you the formula. . Suppose we just wanted to follow 1 rule? We would have 7ways to choose how to do that. . Suppose we wanted to follow 2 rules? There are 7 ways wecould choose to pick the first rule to follow, and then 6 ways tochoose to pick the 2nd rule to follow, for a total of 42 ways topick first one rule and then a second rule. But that assumeswe care about the ORDER in which we pick the rules (thepermutations). Since we don't care about the order, we wantto get rid of all the repeats. That is, if we pick 1 choiceat random from the first 7 and then pick a 2nd choice at randomfrom the remaining 6, we'll end up counting both (smoking,exercise) and (exercise, smoking). In fact, every combinationof two options, option a and option b, will show up in our listtwice, first as (a then b) and second as (b then a). So wecan wipe out all the repeats by dividing by 2. There are 21combinations of ways to choose 2 rules to follow. . (There are also 21 combinations of ways to choose 2 rules NOTto follow, so this actually answers your question.) . The formula for the number of combinations for choosing koptions out of n total choices (n-choose-k) is given by                                     n! / (k!(n-k!)) . So to pick 2 out of 7 we would take 7! / (5!2!) = 7*6*5! /(2*5!) = 7*6/2 = 42/2 = 21 . (The exlamation mark is the mathematical notation for takingthe factorial, that is, n! says to multiply together all theintegers from 1 to n. 3! is 3*2*1; 4! is 4*3*2*1=4*3!) . For part b, you have already chosen 2 rules to follow,so now you need to choose 3 more from the remaining 5 rules.

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