# What is the number of ways letters in the word KOMBINATOORIKA can be rearranged, such that no two consecutive letters are the same

This question is related to my question here. See the following question taken from algebra.com.

What is the number of ways letters in the word KOMBINATOORIKA can be rearranged, such that no two consecutive letters are the same? (the correct answer should be $710579520$, not $100\%$ sure though)

answer provided in www.algebra.com is $498960000$ , screenshot is given below Here is the output of a tool given by careerbless.com [enter the word 'KOMBINATOORIKA', click on generate and answer to the 24th question generated] which gives the final answer as $710579520$ I feel answer is $498960000$ and not $710579520$. For small strings, I had a program using which I was verifying the right answer. But in this case I cannot do it as it is a large string. Could you please help me to find which is the right answer and reason for one answer being wrong.

• Using Jair Taylor's formula also gives 710579520. – user940 Dec 19 '16 at 20:28
• The first method fails at the start since the A's do not need to be separated at that point. Say you have AABMNRT. You can ensure that there are not two consecutive A's by placing a K (or another letter that has not yet been inserted) between them. – N. F. Taussig Dec 19 '16 at 23:26
• is the way the tool solved the problem (second answer) a good method to solve such probolems? – Kiran Dec 20 '16 at 17:15