Neat explanation and answers please

1. Natasha tosses four coins one after the other. 4. a) How many four-digit numbers can you a) In how many different orders could form with the digits 1, 2, 3, 4, 5, 6, and 7 heads or tails occur. if no digit is repeated? b) Draw a tree diagram to illustrate all the b) How many of these four-digit numbers possible results. are odd numbers? c) Explain how your tree diagram c) How many of them are even numbers? corresponds to your calculation in part a). 5. How many ways are there to roll either a 2. Evaluate the following by first expressing 6 or a 12 with two dice? each in terms of factorials. a) $6 b) P(6, 2) c ) P3 6. How many permutations are there of the d) Po e) P(7, 0) letters of each of the following words? a) data b) management c) microwave 3. Suppose you are designing a remote control that uses short, medium, or long pulses of 7. A number of long, thin sticks are lying in a infrared light to send control signals to a pile at odd angles such that the sticks cross device. each other. a) How many different control codes can a) Relate the maximum number of you define using intersection points of n sticks to entries i) three pulses? in Pascal's triangle. ii) one, two, or three pulses? b) What is the maximum number of b) Explain how the multiplicative and intersection points with six overlapping additive counting principles apply in sticks? your calculations for part a)