Problem
COM-B1-M02-P014 Letters \(AABBCC\)
#14
★★★☆☆ Level 3 of 5
How many permutations of \(A,A,B,B,C,C\) have no two equal adjacent letters?
Subtract cases where \(AA\), \(BB\), or \(CC\) forms a block.
Total: \(6!/(2!2!2!)=90\). If \(AA\) is adjacent, count the block \(AA\), \(B,B,C,C\): \(5!/(2!2!)=30\), and there are \(3\) letters. If two pairs are adjacent, for example \(AA\) and \(BB\), we get \(4!/2!=12\), with \(3\) choices. If all three pairs are adjacent, \(3!=6\). Answer: \(90-3\cdot30+3\cdot12-6=30\).
First nontrivial task with overlapping restrictions.