Ravi decides to Choose a 5 digit passcode from the Following 7 characters
1. 9. 6. 7. R. P #
No digits are used twice
A ) how many combinations of 5 digits are there ?
B) Probability that The one he chooses will start with a R and end in P
To find the number of combinations for a 5-digit passcode using 7 characters without repetition, we can use the concept of permutations.
A) Number of Combinations:
Since there are 7 available characters and no digits can be repeated, we need to find the number of ways to arrange these 7 characters in a 5-digit passcode. The formula to calculate this is:
nPr = n! / (n - r)!
where n is the total number of options (characters in this case) and r is the number of selections (5 digits in the passcode). The exclamation mark represents the factorial of a number.
Therefore, for this case, the number of combinations of 5 digits is:
7P5 = 7! / (7 - 5)!
= 7! / 2!
= 7 * 6 * 5 * 4 * 3
= 2,520
So, there are 2,520 possible combinations of 5 digits that can be chosen from the given 7 characters.
B) Probability of Starting with 'R' and Ending with 'P':
To find the probability that the passcode will start with an 'R' and end with a 'P', we need to consider that there is only one 'R' and one 'P' in the given set of characters.
Since we already know that there are 2,520 possible combinations, we need to count the number of combinations where the first digit is 'R' and the last digit is 'P'.
To calculate this, we can treat 'R' and 'P' as fixed positions in the passcode, and the remaining 3 digits can be chosen from the remaining 5 characters (excluding 'R' and 'P'):
3P3 = 3! = 3 * 2 * 1 = 6
So, there are 6 combinations where the passcode will start with 'R' and end with 'P'.
To find the probability, we divide the number of desired outcomes (6) by the total number of possible outcomes (2,520):
Probability = 6 / 2,520 = 1/420
Therefore, the probability that the chosen passcode will start with 'R' and end with 'P' is 1/420.