A factory produces fuses, which are packaged in boxes of 26. Eight fuses are selected at random from each box for inspection. The box is rejected if at least one of these eight fuses is defective. What is the probability that a box containing twelve defective fuses will be rejected?
C (14,8) = 3003
C (26,8) = 1562275
1562275 - 3003 = 1559272
ANSWER= 1559272/1562275 (cannot be simplified)
To find the probability that a box containing twelve defective fuses will be rejected, we need to calculate the probability that none of the eight selected fuses are defective.
First, we need to find the total number of ways to select eight fuses from a box containing 12 defective fuses and 14 non-defective fuses. This can be calculated using combinations:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of fuses and r is the number of fuses selected. In this case, n = 26 (12 defective + 14 non-defective) and r = 8.
C(26, 8) = 26! / (8!(26-8)!)
= 26! / (8!18!)
= (26*25*24*23*22*21*20*19) / (8*7*6*5*4*3*2*1)
Next, we need to find the number of ways to select eight non-defective fuses from 14 non-defective fuses:
C(14, 8) = 14! / (8!(14-8)!)
= 14! / (8!6!)
= (14*13*12*11*10*9*8!) / (8*7*6*5*4*3*2*1)
Finally, we can calculate the probability that none of the eight selected fuses are defective:
P = (number of ways to select eight non-defective fuses) / (total number of ways to select eight fuses)
= C(14, 8) / C(26, 8)
= [(14*13*12*11*10*9*8!) / (8*7*6*5*4*3*2*1)] / [(26*25*24*23*22*21*20*19) / (8*7*6*5*4*3*2*1)]
Simplifying this expression will give us the probability that a box containing twelve defective fuses will be rejected.