[tex] \Large \mathbb{PROBLEM:} [/tex]
[tex] \large \boxed{\:\:\begin{array}{l} \text{How many different five-digit} \\ \text{numbers can be written using} \\ \text{the following digits:} \\ \text{2, 2, 2, 7, 7, 8, 8, 8, and 9?} \end{array}\:\:} [/tex]
[tex] \Large \mathbb{SOLUTION:} [/tex]
[tex] \!\! \begin{array}{l} \large \bold{Given:}\: \{2, 2, 2, 7, 7, 8, 8, 8, 9\} \\ \\ \large \textsf{We must consider 4 cases.} \\ \\ \hline \\ \red{\bold{Case\ 1: \: AAABB}} \\ \\ \textsf{Example: 22277 and its permutations} \\ \\ \textsf{There are only 2 choices for A (either 2 or 8) and} \\ \textsf{3 - 1 = 2 choices for B.} \\ \\ \implies {}^2C_1 \cdot {}^2C_1 \cdot \dfrac{5!}{3!\: 2!} = 2(2)(10) = 40 \\ \\ \hline \\ \red{\bold{Case\ 2:\: AAABC}} \\ \\ \textsf{Example: 22278 and its permutations} \\ \\ \textsf{There are only 2 choices for A (either 2 or 8) and} \\ \textsf{4 - 1 = 3 choices for B and C (so 3 taken 2 at a} \\ \textsf{time).} \\ \\ \implies {}^2C_1 \cdot {}^3C_2 \cdot \dfrac{5!}{3!\: 1!\: 1!} = 2(3)(20) = 120 \\ \\ \hline \\ \red{\bold{Case\ 3:\: AABBC}} \\ \\ \textsf{Example: 22778 and its permutations} \\ \\ \textsf{There are only 3 choices for A and B (so 3 taken} \\ \textsf{2 at a time) and 4 - 2 = 2 choices for C.} \\ \\ \implies {}^3C_2 \cdot {}^2C_1 \cdot \dfrac{5!}{2!\: 2!\: 1!} = 3(2)(30) = 180 \\ \\ \hline \\ \red{\bold{Case\ 4:\: AABCD}} \\ \\ \textsf{Example: 22789 and its permutations} \\ \\ \textsf{There are 3 choices for A (either 2, 7, or 8) and} \\ \textsf{4 - 1 = 3 choices for B, C, and D (so 3 taken 3} \\ \textsf{at a time).} \\ \\ \implies {}^3C_1 \cdot {}^3C_3 \cdot \dfrac{5!}{2!\: 1!\: 1!\: 1!} = 3(1)(60) = 180 \\ \\ \hline \\ \bold{Total}\: = 40 + 120 + 180 + 180 = \boxed{520} \end{array} [/tex]
[tex] \mathfrak{\#Brainlèss\_Squad} [/tex]
