a security alarm requires a four - digit code. the code can use the digits 0 - 9 and the digits cannot be…

a security alarm requires a four - digit code. the code can use the digits 0 - 9 and the digits cannot be repeated. which expression can be used to determine the probability of the alarm code beginning with a number greater than 7?\n\\(\\frac{(_{2}p_{1})(_{9}p_{3})}{_{10}p_{4}}\\)\n\\(\\frac{(_{2}c_{1})(_{9}c_{3})}{_{10}c_{4}}\\)\n\\(\\frac{(_{10}p_{1})(_{9}p_{3})}{_{10}p_{4}}\\)\n\\(\\frac{(_{10}c_{1})(_{9}c_{3})}{_{10}c_{4}}\\)

a security alarm requires a four - digit code. the code can use the digits 0 - 9 and the digits cannot be repeated. which expression can be used to determine the probability of the alarm code beginning with a number greater than 7?\n\\(\\frac{(_{2}p_{1})(_{9}p_{3})}{_{10}p_{4}}\\)\n\\(\\frac{(_{2}c_{1})(_{9}c_{3})}{_{10}c_{4}}\\)\n\\(\\frac{(_{10}p_{1})(_{9}p_{3})}{_{10}p_{4}}\\)\n\\(\\frac{(_{10}c_{1})(_{9}c_{3})}{_{10}c_{4}}\\)

Answer

Explanation:

Step1: Calculate total number of 4 - digit codes

The total number of four - digit codes using digits 0 - 9 without repetition is given by the permutation formula ({n}P{r}=\frac{n!}{(n - r)!}), where (n = 10) and (r=4). So the total number of codes is ({10}P{4}).

Step2: Calculate number of 4 - digit codes starting with a number greater than 7

Numbers greater than 7 in the set ({0,1,\cdots,9}) are 8 and 9. So there are 2 choices for the first digit (({2}P{1})). After choosing the first - digit, we need to choose the remaining 3 digits from the remaining 9 digits. The number of ways to choose and arrange the remaining 3 digits is ({9}P{3}). The number of 4 - digit codes starting with a number greater than 7 is (({2}P{1})({9}P{3})).

Step3: Calculate probability

Probability (P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}). So the probability of the alarm code beginning with a number greater than 7 is (\frac{({2}P{1})({9}P{3})}{{10}P{4}}).

Answer:

(\frac{({2}P{1})({9}P{3})}{{10}P{4}})