Brittany is going to ascend a six-step staircase. At any time, she is just as likely to stride up one step or three steps. Find the probability of the event that she will ascend the six steps in the following number of strides. \( \begin{array}{ll}\text { a. two strides } & \text { b. four strides } \\ \text { c. six strides }\end{array} \) \( \begin{array}{l}\text { a. The probability that Brittany will ascend the staircase in two strides is } \frac{1}{4}\end{array} \) (Type an integer or a simplified fraction.) b. The probability that Brittany will ascend the staircase in four strides is \( \frac{11}{16} \) (Type an integer or a simplified fraction.) c. The probability that Brittany will ascend the staircase in six strides is (Type an integer or a simplified fraction.)

To solve the problem, we need to analyze how Brittany can ascend the six-step staircase using strides of either one step or three steps. We'll calculate the probabilities for each case: two strides, four strides, and six strides. ### a. Two Strides To ascend the staircase in two strides, Brittany must take one stride of three steps and one stride of three steps. The only combination is: - Stride of 3 steps + Stride of 3 steps Thus, there is only one way to achieve this. The total number of ways to take two strides (either 1 step or 3 steps) can be calculated as follows: 1. Each stride can be either 1 step or 3 steps. 2. The total number of combinations for two strides is \(2^2 = 4\). The probability is then calculated as: \[ P(\text{2 strides}) = \frac{\text{Number of successful outcomes}}{\text{Total outcomes}} = \frac{1}{4} \] ### b. Four Strides To ascend the staircase in four strides, we can have different combinations of 1-step and 3-step strides. Let \(x\) be the number of 1-step strides and \(y\) be the number of 3-step strides. We need to satisfy the following equations: 1. \(x + y = 4\) (total number of strides) 2. \(x + 3y = 6\) (total steps) From the first equation, we can express \(x\) in terms of \(y\): \[ x = 4 - y \] Substituting into the second equation: \[ (4 - y) + 3y = 6 \implies 4 + 2y = 6 \implies 2y = 2 \implies y = 1 \] Thus, \(x = 4 - 1 = 3\). The valid combination is: - 3 strides of 1 step and 1 stride of 3 steps. The number of ways to arrange these strides is given by the multinomial coefficient: \[ \text{Number of ways} = \frac{4!}{3!1!} = 4 \] The total number of ways to take four strides is \(2^4 = 16\). Thus, the probability is: \[ P(\text{4 strides}) = \frac{4}{16} = \frac{1}{4} \] ### c. Six Strides To ascend the staircase in six strides, Brittany can only take 1-step strides since taking a 3-step stride would exceed the total of 6 steps. Therefore, the only combination is: - 6 strides of 1 step. The total number of ways to take six strides is \(2^6 = 64\). Thus, the probability is: \[ P(\text{6 strides}) = \frac{1}{64} \] ### Summary of Probabilities - a. The probability that Brittany will ascend the staircase in two strides is \( \frac{1}{4} \). - b. The probability that Brittany will ascend the staircase in four strides is \( \frac{1}{4} \). - c. The probability that Brittany will ascend the staircase in six strides is \( \frac{1}{64} \).