SOLUTION: Your bank requires you to pick a four digit PIN for your debit card from the digits 0-9. How many different PINs are possible if adjacent digits cannot be the same? (For example th

First let's count all possible types of unacceptable PINs, and then
count the number in each type

There are 3 types of unacceptable PINs which use exactly 3 different digits

AABC, for instance 7792
ABBC, for instance 4885
ABCC, for instance 6099

Those account for 10·9·8 each or 3·10·9·8 = 2160

There are 6 types which use exactly 2 different digits

AAAB, for instance 5551
AABA, for instance 3373
ABAA, for instance 5355
BAAA, for instance 7333
AABB, for instance 6611
ABBA, for instance 2882


Those account for 10·9 each or 6·10·9 = 540

There is only 1 type that uses exactly 1 digit.

AAAA, for instance 7777

There are 10 of those.

So we have a grand total of 2160+540+10 = 2710 unacceptable PINs

There are 10·10·10·10 or 10000 4 digit PINs with no restrictions.

So there are 10000-2710 or 7290 acceptable PINs.

Answer: 7290

Edwin