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I am trying to reconcile two approaches used in counting problems. The first approach uses combinations and the other uses probability. I understand the combinations approach, but not able to comprehend the probability approach. Consider the following example,

A carton contains 12 toasters, 3 of which are defective (D). If four toasters are sold at random, find the probability that exactly one will be defective.

By the combinations approach: 3C1 * 9C3 / 12C4 = 252/495 = 0.509

Now the probability approach:

I am purposely want to use here permutations and not combinations.

total possible permutations = 12P4 =11880

Another way of finding the total permutations is to find the chance of getting any permutation = 1/12 * 1/11* 1/10*1/9 = 0.000084. The reciprocal of this number gives the number 11880.

My confusion starts now. Consider the case DGGG, where G = Good, and D= Defective.

P (DGGG) = 3/12 * 9/11 * 8/10 * 7/9 = 0.127

If we divide this number by the probability of each permutation, 0.000084, we get 1512.

Thus, DGGG accounts for 1512 permutations or 1512 / 24 = 63 combinations.

I am having a hard time understanding how these 1512 permutations or 63 combinations look like.

Could you please help me with this?

Thanks,

MG.

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# Homework Help: Counting using probability

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