20. Frederik wants to make a special seven-digit password. Each digit of his password occurs exactly as many times as its digit value. The digits with equal values always occur consecutively, e.g. 4444333 or 1666666 . How many possible passwords can he make?
A 6
В 7
C 10
D 12
E 13
20. E The password has length 7 so the different digits making it up must add to 7. The possibilities are: $\{7\},\{6,1\},\{5,2\},\{4,3\},\{4,2,1\}$. Using only the digit 7 produces just one password, 7777777. Using two digits gives two possibilities, depending on which digit goes first, so the three pairs give $2 \times 3=6$ phone numbers. Three different digits can be arranged in six ways. This gives $1+6+6=13$ possibilities.