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How many five-digit positive integers can be written using only the digits 1,2 and 3 and do not have two consecutive digits which are both 3 's? (For example, 32322  satisfies these conditions, while 33132 does not.)

 Jul 20, 2022
 #1
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(11111, 11112, 11113, 11121, 11122, 11123, 11131, 11132, 11211, 11212, 11213, 11221, 11222, 11223, 11231, 11232, 11311, 11312, 11313, 11321, 11322, 11323, 12111, 12112, 12113, 12121, 12122, 12123, 12131, 12132, 12211, 12212, 12213, 12221, 12222, 12223, 12231, 12232, 12311, 12312, 12313, 12321, 12322, 12323, 13111, 13112, 13113, 13121, 13122, 13123, 13131, 13132, 13211, 13212, 13213, 13221, 13222, 13223, 13231, 13232, 21111, 21112, 21113, 21121, 21122, 21123, 21131, 21132, 21211, 21212, 21213, 21221, 21222, 21223, 21231, 21232, 21311, 21312, 21313, 21321, 21322, 21323, 22111, 22112, 22113, 22121, 22122, 22123, 22131, 22132, 22211, 22212, 22213, 22221, 22222, 22223, 22231, 22232, 22311, 22312, 22313, 22321, 22322, 22323, 23111, 23112, 23113, 23121, 23122, 23123, 23131, 23132, 23211, 23212, 23213, 23221, 23222, 23223, 23231, 23232, 31111, 31112, 31113, 31121, 31122, 31123, 31131, 31132, 31211, 31212, 31213, 31221, 31222, 31223, 31231, 31232, 31311, 31312, 31313, 31321, 31322, 31323, 32111, 32112, 32113, 32121, 32122, 32123, 32131, 32132, 32211, 32212, 32213, 32221, 32222, 32223, 32231, 32232, 32311, 32312, 32313, 32321, 32322, 32323)>>Total == 164 such permutations.

 Jul 20, 2022

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