Counting with digit/number constraints

Count how many numbers can be formed from a set of digits subject to constraints like being even, being in a certain range, having specific digits in specific positions, or using distinct digits.

1 questions · Moderate -0.8

5.01a Permutations and combinations: evaluate probabilities
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OCR Further Discrete AS 2019 June Q1
5 marks Moderate -0.8
1 Alfie has a set of 15 cards numbered consecutively from 1 to 15.
He chooses two of the cards.
  1. How many different sets of two cards are possible? Alfie places the two cards side by side to form a number with 2,3 or 4 digits.
  2. Explain why there are fewer than \({ } ^ { 15 } \mathrm { P } _ { 2 } = 210\) possible numbers that can be made.
  3. Explain why, with these cards, 1 is the lead digit more often than any other digit. Alfie makes the number 113, which is a 3-digit prime number. Alfie says that the problem of working out how many 3-digit prime numbers can be made using two of the cards is a construction problem, because he is trying to find all of them.
  4. Explain why Alfie is wrong to say this is a construction problem.