Question

How many selections of four digits from 0,…,9 are there such that the following hold? Note, to type in (nr​) you type in binomial (n,r). a.) No digit is repeated. Your last answer was interpreted as follows: (94​) . b). No digit is repeated, and the digits must be in ascending order. Your last answer was interpreted as follows: (49 ). c). No digit is repeated, and the third digit must be a zero.

Answer 1

The number of selections of four digits from 0 to 9 without repeating is 10P4, in ascending order is (10C4), and with the third digit being zero is (9C3).

Step-by-step explanation:

To determine how many selections of four digits from 0 to 9 can be made under different conditions:

1. No digit is repeated: We use the formula for permutations of selecting 4 digits from 10, without repetition, which is 10P4 or (10!)/(6!).
2. No digit is repeated, and the digits must be in ascending order: The number of combinations is given by the binomial coefficient (10C4) or binomial(10,4) since the order does not matter when they are in ascending order.
3. No digit is repeated, and the third digit must be a zero: Since the third digit is fixed as zero, we choose 3 more digits from the remaining 9 digits, which gives us (9C3) or binomial(9,3) combinations.