How many 2-digit numbers can be formed using only the digits 2, 3, 5, and 6, if the digits are not be repeated within a number?
A. 11
B. 12
C. 10
D. 2
There are 4 choices for the tens digit and 3 choices for the units digit (since it cannot be the same as the tens digit). Therefore, there are $4\times 3 = 12$ such 2-digit numbers. The answer is $\boxed{\textbf{(B)}\ 12}$.
A juice company decides to test five different brands of juice. The different brands have been labeled A¸ B¸ C¸ D¸ and E. The company decides to compare each brand with the other brands by pairing together different brands. How many different pairs will result by selecting two different brands at a time?
A. 11
B. 15
C. 120
D. 10
We want to count the number of ways to choose 2 brands out of 5, which is just $\binom{5}{2}=\boxed{\textbf{(B)}\ 15}$.
To find the number of 2-digit numbers that can be formed using only the digits 2, 3, 5, and 6 without repeating any digits, we need to consider two cases: one where the first digit is 2 and one where the first digit is not 2.
Case 1: The first digit is 2.
In this case, we can choose the second digit from the remaining 3 digits (3, 5, and 6). So there are 3 choices for the second digit.
Case 2: The first digit is not 2.
In this case, we can choose the first digit from the remaining 3 digits (3, 5, and 6). After choosing the first digit, we can then choose the second digit from the remaining 2 digits. So there are 3 choices for the first digit and 2 choices for the second digit, resulting in a total of 3 * 2 = 6 choices.
Therefore, the total number of 2-digit numbers that can be formed is 3 + 6 = 9.
Thus, the correct answer is not listed among the provided options.