Permutations and Combinations
3.7 Yet More Practice
All solutions appear in the COMPANION GUIDE to this Permutations and Combinations course.
Exercise 39: Choose any entry in Pascal’s triangle – say the “15” circled in the diagram below. Form the parallelogram of entries above it just as shown.
Sum all the entries in the parallelogram. (In this case we have: 15 + 10 + 5 + 6 + 4 + 1 + 3 + 3 + 1 + 1 + 2 + 1 + 1 + 1 + 1.) The answer is one less than the value of the entry directly below the circled entry two rows down. (In this case, one less than 56.)
a) Play with this. Can you see why 56 is one more than the sum of entries in the parallelogram? (A yes/no answer will suffice.) HINT: Think of the entries in the parallelogram as a series of parallel stockings.
b) Can you see that this parallelogram property always works? (Again an honest yes or no will suffice.) 
Exercise 40: Suppose we wish to expand the product \(\left(x+y+z\right)^{7}\).
a) How many times will the term \(y^{7}\) appear? b) How many times will the term \(xz^{6}\) appear? c) How many times will the term \(x^{2}y^{3}z^{2}\) appear? d) Write a formula for the number of times a general term \(x^{a}y^{b}z^{c}\) will appear. e) State the general “trinomial theorem:” \(\left(x+y+z\right)^{n}=\) f) State a general “quadronomial” theorem: \(\left(x+y+z+w\right)^{n}=\)

Exercise 41: Sally has a large collection of \(1 \times 1\), \(1 \times 2\), \(1 \times 3\), \(1 \times 4\), … blocks, an infinite number of each type in fact. With these blocks she likes to build “trains,” which consist of rows of blocks. Sally noticed that she can make eight essentially different trains of length four:
She also noticed that there is 1 train using four blocks, 3 using three blocks, 3 using two blocks and 1 using one block.
a) Draw all the trains of length five Sally can make. Group them according to the number of blocks in each train.
b) Why the connection to Pascal’s triangle? 
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