Competition problem — Combinatorics

Library / /273 of 520

Combinatorics Difficulty 5.6 AIME, harder Find the answer

8. In the six-digit decimal number $\overline{a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}}$ , each digit $a_{i}(1 \leqslant i \leqslant 6)$ is an odd number, and the digit 1 is not allowed to be adjacent (for example, 135131, 577797 satisfy the condition, while 311533 does not satisfy the condition), then the number of such six-digit numbers is $\qquad$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

When $a_{i}(1 \leqslant i \leqslant 6)$ does not contain the digit 1, the number of six-digit numbers $\overline{a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}}$ is $4^{6}$ ; when $a_{i}(1 \leqslant i \leqslant 6)$ contains one 1, the number of six-digit numbers $\overline{a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}}$ is $6 \cdot 4^{5}$ ; when $a_{i}(1 \leqslant i \leqslant 6)$ contains two 1s, the number of six-digit numbers $\overline{a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}}$ is $C_{5}^{2} \cdot 4^{4}$ ; when $a_{i}(1 \leqslant i \leqslant 6)$ contains three 1s, the number of six-digit numbers $\overline{a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}}$ is $C_{4}^{3} \cdot 4^{3}$ , so the number of six-digit numbers $\overline{a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}}$ that satisfy the condition is 13056.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.