cecile tosses 5 coins one after another
a. how many different outcomes are possible
b. draw a tree to illustrate the different possibilities
c. in how many ways will the first coin turn up heads and the last coin turn up tails
d. in how many ways will the second and thrd and fourth turn up all heads?
a. To find the number of different outcomes possible when tossing 5 coins, we need to consider the number of possibilities for each coin toss. Since each coin has 2 possible outcomes (heads or tails), the total number of outcomes can be calculated by multiplying the number of possibilities for each toss together: 2 * 2 * 2 * 2 * 2 = 2^5 = 32 different outcomes are possible.
b. Here is a tree diagram illustrating the different possibilities for tossing 5 coins:
_____ H _____ H _____ H _____ H _____ H
/
H _____ H _____ H _____ H _____ T
/ \
/ \
H _____ H _____ H _____ T T _____ H _____ H _____ H _____ T
\ /
\ /
H _____ H _____ T _____ T _____ H
\
\_____ T _____ T _____ T _____ T
In this tree diagram, each branch represents a coin toss, and "H" and "T" represent the possible outcomes (heads and tails).
c. Since we want the first coin to turn up heads and the last coin to turn up tails, we can treat these outcomes as fixed. Therefore, we only need to consider the possibilities for the remaining 3 coins. Applying the formula from part a, we have 2 * 2 * 2 = 2^3 = 8 different ways the second, third, and fourth coins can turn up. However, since the first and last coins are already determined, the total number of ways is 8 * 1 * 1 = 8.
d. To have the second, third, and fourth coins all turn up heads, we can treat these outcomes as fixed. Therefore, we only need to consider the possibilities for the remaining 2 coins (first and last). We know that the first coin must be tails and the last coin must be heads. Therefore, there are only 1 possible outcome for each of those coins. Applying the formula from part a, we have 1 * 1 * 1 = 1^3 = 1 different way for the second, third, and fourth coins to all turn up heads.