61 lines
2.9 KiB
XML
61 lines
2.9 KiB
XML
Section 6.4: 7, *9*, *11*, 17, *21*, 23, *29, 33, 35, 37, 39*
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= Exercise 7
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What is the coefficient of $x⁹ "in" (2-x)^(19)$
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= Exercise 9
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What is the coefficient of $x^101y^99 "in the expansion of" (2x-3y)^200$
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*Answer:*
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$ -200!/(101! dot 99!)dot 2^101 dot 3^99 = -2^101 dot 3^99 dot mat(200;99) $
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= Exercise 11
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Use the binomial theorem to expand $(3x⁴-2y³)⁵$ into a sum of terms of the form $c x^a y^b$, where $c$ is a real number and $a$ and $b$ are nonnegative integers
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*Answer:*
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$ sum^5_(k=0)mat(5;k)(3x^4)^(5-k) (-2y^3)^(k) =\
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243x^20-81dot x^16dot 2y^3 dot 5 + 27 dot x^12dot 4y^6dot 10 - 9x^8dot 8y^9dot 10 + 3x^4 dot 16y^12dot 5 -32y^15 =\
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243x^20-810x^16y^3+1080x^12y^6-720x^8y^9+240x^4y^12-32y^15 $
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= Exercise 17
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What is the row of Pascal's triangle containing the binomial coefficients $mat(9;k),0<=k<=9$
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= Exercise 21
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Show that if $n$ and $k$ are integers with $1<=k<=n$, then $mat(n;k)<=n^k / 2^(k-1)$
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*Answer:*
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We know that $mat(n;k)$ is $n!/((n-k)!k!) = (n(n-1)(n-2)dots n-k+1)/(k(k-1)(k-2)(k-3)dots 2) <= (n dot n dot n dot dots)/(2 dot 2 dot 2)$
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= Exercise 23
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Prove Pascal's identity, using the formula for $mat(n;r)$
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= Exercise 29
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Let $n$ be a positive integer. Show that $ mat(2n;n+1)+mat(2n;n) = mat(2n+2;n+1) / 2 $
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*Answer:*
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We know:
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$mat(2n;n+1)+mat(2n;n)=mat(2n+1;n+1)$
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$mat(2n+1;n+1)=1/2dot (mat(2n+1;n+1)mat(2n+1;n+1))=mat(2n+2;n+1)/2$
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= Exercise 33
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Give a combinatorial proof that $sum_(k=1)^n k mat(n;k)=n 2^(n-1)$.
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_Hint: Count in two ways the number of ways to select a committee and to then select a leader of the committee._
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*Answer:*
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To select a committee, you have $2^(n-1)$ choices (n-1 because when you have n=1, then the choices are $2^0=1$) and then you select one from the `n` people in the committee.
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= Exercise 35
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Show that a nonempty set has the same number of subsets with an odd number of elements as it does subsets with an even number of elements
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= Exercise 37
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In this exercise we will count the number of paths in the $x y$ plane between the origin $(0,0)$ and point $(m,n)$, where $m$ and $n$ are nonnegative integers, such that each path is made up of a series of steps, where each step is a move unit to the right or a move unit upward. (No moves to the left or downward are allowed.) Two such paths from $(0,0)" to "(5,3)$ are illustrated here.
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#image("Exercise 6.4-37.png")
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a) Show that each path of the type described can be represented by a bit string consisting of $m$ 0s and $n$ 1s, where a 0 represents a move one unit to the right and a 1 represents a move one unit upward.
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b) Conclude from part (a) that there are $mat(m+n;n)$ paths of the desired type
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= Exercise 39
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Use Exercise 37 to prove Theorem 4. _Hint: Count the number of paths with $n$ steps of the type described in Exercise 37. Every such path must end at one of the points $(n-k,k)$ for $k=0,1,2, dots, n$._
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Theorem 4: Let $n$ and $r$ be nonnegative integers with $r<=n$. Then: $mat(n+1;r+1)=sum^n_(j=r) mat(j;r)$ |