5. Matthew has black socks white socks dress pants, a red shirt, green shirt, and a T-shirt. a) Draw a tree diagram showing his choices for socks, pants , and shirt b) Find the probability that Matthew selects at random i) blue jeans and the T-shirt ii) white socks iii) black socks, dress pants, and a red shirt iv) white socks and not the T -shirt

## Step 1: Create the Tree Diagram<br />### Matthew has two choices for socks (black or white), one choice for pants (dress pants), and three choices for shirts (red, green, or T-shirt). We can represent this with a tree diagram. The first branch represents socks, the second branch represents pants, and the third branch represents shirts.<br /><br />## Step 2: Calculate Total Outcomes<br />### Each path from the beginning to the end of the tree diagram represents a possible outcome. There are $2 \times 1 \times 3 = 6$ total possible outcomes.<br /><br />## Step 3: Calculate Probability of Blue Jeans and T-shirt<br />### The problem states Matthew has dress pants, not blue jeans. Therefore, the probability of choosing blue jeans and a T-shirt is 0.<br /><br />## Step 4: Calculate Probability of White Socks<br />### There are 3 outcomes with white socks out of 6 total outcomes. The probability is $\frac{3}{6} = \frac{1}{2}$.<br /><br />## Step 5: Calculate Probability of Black Socks, Dress Pants, and Red Shirt<br />### There is 1 outcome with black socks, dress pants, and a red shirt out of 6 total outcomes. The probability is $\frac{1}{6}$.<br /><br />## Step 6: Calculate Probability of White Socks and Not a T-shirt<br />### There are 2 outcomes with white socks and not a T-shirt (white socks with red shirt and white socks with green shirt) out of 6 total outcomes. The probability is $\frac{2}{6} = \frac{1}{3}$.