analyzing scenarios to calculate probabilities\na bicycle lock has a four - digit code. the possible digits…

Explanation:

Step1: Calculate total number of four - digit codes

The number of permutations of (n) distinct objects taken (r) at a time is given by (P(n,r)=\frac{n!}{(n - r)!}). Here, (n = 10) (digits from 0 - 9) and (r=4), so the total number of four - digit codes with no repeated digits is (P(10,4)=\frac{10!}{(10 - 4)!}=\frac{10!}{6!}=10\times9\times8\times7 = 5040).

Step2: Calculate number of codes starting with 5

If the code starts with 5, then we need to arrange the remaining 9 digits in the remaining 3 positions. So the number of such codes is (P(9,3)=\frac{9!}{(9 - 3)!}=\frac{9!}{6!}=9\times8\times7=504). The probability that the lock code begins with 5 is (\frac{504}{5040}=\frac{1}{10}).

Step3: Calculate number of codes without 0

If the code does not contain 0, then we are choosing 4 digits out of 9 non - zero digits. The number of such codes is (P(9,4)=\frac{9!}{(9 - 4)!}=\frac{9!}{5!}=9\times8\times7\times6 = 3024). The probability that the lock code does not contain 0 is (\frac{3024}{5040}=\frac{3}{5}).

Answer:

The probability that the lock code will begin with 5 is (\frac{1}{10}). The probability that the lock code will not contain the number 0 is (\frac{3}{5}).