In (a, b, c, d, e, f, g, h, 1, 2, 3, 4), how many subsets contain all four numerals?

Question

How abt in (a, b, c, d, e, f, g, h, 1, 2, 3, 4), how many subsets contain all four numerals?

How many contain exactly two numerals?

How many contain exactly 3 letters and two numerals?

Answer 1

For a subset to contain all four numerals, it must contain 1, 2, 3 and 4 (obviously) and can contain any combination of the other 8 elements. The number of subsets of {a, b, c, d, e, f, g, h} is 2^8 = 256, so there are 256 subsets of the original set containing all four numerals.

To contain exactly two numerals, we again have 2^8 = 256 ways of choosing the letter part, and we have 4 choose 2 = (4.3)/(2.1) = 6 ways of choosing the two numerals. So there are 256 * 6 = 1536 such subsets in all.

To contain exactly 3 letters and 2 numerals, we have (8 choose 3) * (4 choose 2) = (8.7.6)/(3.2.1) . (4.3)/(2.1) = 56 . 6 = 336 subsets.

Answer 2

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