<p>This question came of College Board's 10 real SATs:</p>

<p>What is the number of different numbers that can be formed by rearranging the digits in the number 2,224, keeping 2 in the thousands place?</p>

<p>I know that this can be done easily enough by just listing, but hypothetically, if there were a question that would take too much time to list systematically, is there a formula I could use to solve this exact question?</p>

<p>As you have noted, inspection probably is the fastest method:</p>

<p>First digit must be a "2" so the "4" can go in the hundreds, tens or units spot yielding three distinguishable numbers.</p>

<p>The relevant formula for permutations of with some indistinguishable items is n!/m! where n is the number of items to be permuted and m is the number of those items which are identical; in this case, we would again put a "2" in the thousands place and permute 2-2-4 (three items, two identical) to find 3!/2! (=3) distinct permutations.</p>