$32 \cdot 19$ Among all six-digit numbers, the sum of the digits equals 52. How many such six-digit numbers are there?
(A) 2.
(B) 12.
(C) 21.
(D) 31.
(China Zhejiang Province Junior High School Mathematics Competition, 1991)
[Solution] The largest six-digit number is 999999, with the sum of its digits being 54. Therefore, a six-digit number with a digit sum of 52 can only have (1) five 9s and one 7, or (2) four 9s and two 8s.
The digits $9,9,9,9,9,7$ can form 6 different six-digit numbers;
The digits $9,9,9,9,8,8$ can form 15 different six-digit numbers (the case where the two 8s are together has 5 possibilities, the case where the two 8s are separated by two 9s has 3 possibilities, the case where the two 8s are separated by one or three 9s has 6 possibilities, and the case where the two 8s are separated by four 9s has 1 possibility).
Furthermore, there are no other scenarios where the sum of the six digits is 52.
Therefore, the total number of six-digit numbers is $6+15=21$.
Hence, the answer is $(C)$.