Which of the following are countable is point on line}

A set is called countable if it is either finite, or if it is infinite but its elements can be listed in a sequence that matches one-to-one with the natural numbers (a countably infinite set). Even a single determinate quantity — viewed as a one-element set — is finite, and therefore automatically countable. A set that admits no such one-to-one listing at all — such as the set of real numbers or the continuum of points on a line — is uncountable.

• I: A is the set of points on a line. Because the points on a line form a continuum (there are uncountably many points between any two of them), no sequence can list every point in one-to-one correspondence with the natural numbers — by Cantor's diagonal argument, this set is uncountable.

• II: B = {x : x ∈ N and x < 100} = {1, 2, …, 99} is a finite set with only 99 elements. Every finite set is trivially countable, since its elements can simply be listed one by one.

• III: C is the number of permutations of the letters of the largest possible English word — a single, specific value once that word is fixed. Since any English word has a finite number of letters, that permutation count is one determinate finite number, and any determinate finite quantity is (trivially) countable.

The distinguishing test is whether a set's elements can be put in one-to-one correspondence with a subset of the natural numbers. B and C reduce to finite collections, which always pass this test; A does not, because between any two points on a line lie infinitely many further points with no smallest gap — matching the classic uncountability of the real line.

So only II and III are countable, while I is not.