From 861062fb68c32fb7c25177107edcd9b61f72519c Mon Sep 17 00:00:00 2001 From: aidan <144724440+aidanln@users.noreply.github.com> Date: Mon, 16 Dec 2024 20:10:40 -0500 Subject: [PATCH 1/5] added f2024 fall exam category, fixed w2022 final q6 solution --- .../evaluations/comp2804/2024-fall-final.md | 33 +++++++++++++++++++ .../comp2804/2022-winter-final/6/solution.md | 12 +++---- 2 files changed, 39 insertions(+), 6 deletions(-) create mode 100644 src/content/evaluations/comp2804/2024-fall-final.md diff --git a/src/content/evaluations/comp2804/2024-fall-final.md b/src/content/evaluations/comp2804/2024-fall-final.md new file mode 100644 index 00000000..c248fa6b --- /dev/null +++ b/src/content/evaluations/comp2804/2024-fall-final.md @@ -0,0 +1,33 @@ +--- +title: 2024 Fall Final +path: comp2804/2024-fall-final +author: Patrick Morin +course: comp2804 +created_at: 2024-12-16 +questions: + - comp2804/2024-fall-final/1 + - comp2804/2024-fall-final/2 + - comp2804/2024-fall-final/3 + - comp2804/2024-fall-final/4 + - comp2804/2024-fall-final/5 + - comp2804/2024-fall-final/6 + - comp2804/2024-fall-final/7 + - comp2804/2024-fall-final/8 + - comp2804/2024-fall-final/9 + - comp2804/2024-fall-final/10 + - comp2804/2024-fall-final/11 + - comp2804/2024-fall-final/12 + - comp2804/2024-fall-final/13 + - comp2804/2024-fall-final/14 + - comp2804/2024-fall-final/15 + - comp2804/2024-fall-final/16 + - comp2804/2024-fall-final/17 + - comp2804/2024-fall-final/18 + - comp2804/2024-fall-final/19 + - comp2804/2024-fall-final/20 + - comp2804/2024-fall-final/21 + - comp2804/2024-fall-final/22 + - comp2804/2024-fall-final/23 + - comp2804/2024-fall-final/24 + - comp2804/2024-fall-final/25 +--- diff --git a/src/content/questions/comp2804/2022-winter-final/6/solution.md b/src/content/questions/comp2804/2022-winter-final/6/solution.md index ebe67177..77dc8c32 100644 --- a/src/content/questions/comp2804/2022-winter-final/6/solution.md +++ b/src/content/questions/comp2804/2022-winter-final/6/solution.md @@ -1,13 +1,13 @@ $ (2x - 3y)^{30} $ -$ = \sum\_{k=10}^{30} \binom{30}{k} \cdot (2x)^{k} \cdot (-3y)^{30-k} $ +$ = \sum\_{k=20}^{30} \binom{30}{k} \cdot (2x)^{30-k} \cdot (-3y)^{k} $ -$ = \binom{30}{10} \cdot (2x)^{10} \cdot (-3y)^{30-10} $ +$ = \binom{30}{20} \cdot (2x)^{30-20} \cdot (-3y)^{20} $ -$ = \binom{30}{10} \cdot (2x)^{10} \cdot (-3y)^{20} $ +$ = \binom{30}{20} \cdot (2x)^{10} \cdot (-3y)^{20} $ -$ = \binom{30}{10} \cdot 2^{10} \cdot (-3)^{20} \cdot x^{10} \cdot y^{20} $ +$ = \binom{30}{20} \cdot 2^{10} \cdot (-3)^{20} \cdot x^{10} \cdot y^{20} $ -$ = \binom{30}{10} \cdot 2^{10} \cdot (3)^{20} \cdot x^{10} \cdot y^{20} $ +$ = \binom{30}{20} \cdot 2^{10} \cdot (3)^{20} \cdot x^{10} \cdot y^{20} $ -From this equation, we can see that the coefficient (aka the real numbers) are: $\binom{30}{10} \cdot 2^{10} \cdot (3)^{20}$ +From this equation, we can see that the coefficient (aka the real numbers) are: $\binom{30}{20} \cdot 2^{10} \cdot 3^{20}$ From cd82b3fb7e2240ea68760d326a420764597a436b Mon Sep 17 00:00:00 2001 From: aidan <144724440+aidanln@users.noreply.github.com> Date: Mon, 16 Dec 2024 21:44:39 -0500 Subject: [PATCH 2/5] improved NBT solutions, fixed an incorrect tag --- .../comp2804/2014-fall-midterm/7/solution.md | 10 +++++----- .../questions/comp2804/2015-fall-final/4/solution.md | 12 ++++++++---- .../questions/comp2804/2015-fall-midterm/7/index.md | 2 +- .../comp2804/2015-winter-final/4/solution.md | 12 +++++++++--- .../comp2804/2016-fall-midterm/9/solution.md | 12 ++++++++---- .../comp2804/2017-fall-midterm/9/solution.md | 12 ++++++------ .../questions/comp2804/2018-fall-final/6/solution.md | 12 ++++++++---- .../comp2804/2022-winter-final/6/solution.md | 4 +++- 8 files changed, 48 insertions(+), 28 deletions(-) diff --git a/src/content/questions/comp2804/2014-fall-midterm/7/solution.md b/src/content/questions/comp2804/2014-fall-midterm/7/solution.md index 42878a37..31fcbe69 100644 --- a/src/content/questions/comp2804/2014-fall-midterm/7/solution.md +++ b/src/content/questions/comp2804/2014-fall-midterm/7/solution.md @@ -1,11 +1,11 @@ ${(5x-36)}^{100}$ -$=\sum^{100}_{k=0} \binom{100}{k}{(5x)}^k {(-3y)}^{n-k}$ +$ = \sum^{100}_{k=0} \binom{100}{k} {(5x)}^{n-k} {(-3y)}^{k} $ -$=\binom{100}{20}{(5x)}^{20} {(-3y)}^{80}$ +We only consider $k=80$, as it results in $y^{80}$. -$=\binom{100}{20}5^{20} 3^{80} x^{20} y^{80}$ +$ = \binom{100}{80} \cdot {(5x)}^{100-80} \cdot {(-3y)}^{80} $ -$=\binom{100}{80}5^{20} 3^{80} x^{20} y^{80}$ +$ = \binom{100}{80} \cdot 5^{20} \cdot {(-3)}^{80} \cdot x^{20} \cdot y^{80} $ -$=\binom{100}{80}5^{20} 3^{80}$ (this is the coefficient) +$ = \binom{100}{80} \cdot 5^{20} \cdot 3^{80} $ (final answer, i.e. the coefficient of $x^{20} y^{80}$) diff --git a/src/content/questions/comp2804/2015-fall-final/4/solution.md b/src/content/questions/comp2804/2015-fall-final/4/solution.md index 3a0f9a40..fb663a23 100644 --- a/src/content/questions/comp2804/2015-fall-final/4/solution.md +++ b/src/content/questions/comp2804/2015-fall-final/4/solution.md @@ -1,7 +1,11 @@ -$ = \sum^{20}\_{k=0} \binom{20}{k} {(-3x)}^{k} {(5y)}^{20-k} $ +$ = \sum^{20}_{k=0} \binom{20}{k} {(-3x)}^{n-k} {(5y)}^{k} $ -$ = \binom{20}{15} {(-3)}^{15} {5}^{5} x^{15} y^5 $ +We only consider $k=5$, as it results in $y^{5}$. -$ = - \binom{20}{15} {(3)}^{15} {5}^{5} x^{15} y^5 $ +$ = \binom{20}{5} \cdot {(-3x)}^{20-5} \cdot {(5y)}^{5} $ -Thus, the coefficient of $x^{15}y^{5}$ in the expansion of ${(-3x + 5y)}^{20}$ is $ - \binom{20}{15} {(3)}^{15} {5}^{5} $ +$ = \binom{20}{5} \cdot {(-3)}^{15} \cdot {5}^{5} \cdot x^{15} \cdot y^5 $ + +$ = - \binom{20}{5} \cdot {3}^{15} \cdot {5}^{5} \cdot x^{15} \cdot y^5 $ + +Thus, the coefficient of $x^{15}y^{5}$ in the expansion of ${(-3x + 5y)}^{20}$ is $ - \binom{20}{5} \cdot {3}^{15} \cdot {5}^{5} $ diff --git a/src/content/questions/comp2804/2015-fall-midterm/7/index.md b/src/content/questions/comp2804/2015-fall-midterm/7/index.md index cba52a19..cf0cef72 100644 --- a/src/content/questions/comp2804/2015-fall-midterm/7/index.md +++ b/src/content/questions/comp2804/2015-fall-midterm/7/index.md @@ -8,5 +8,5 @@ solution: comp2804/2015-fall-midterm/7/solution.md tags: - comp2804 - comp2804-midterm - - comp2804-newton's-binomial-theorem + - comp2804-the-pigeonhole-principle --- diff --git a/src/content/questions/comp2804/2015-winter-final/4/solution.md b/src/content/questions/comp2804/2015-winter-final/4/solution.md index dd57f448..34455183 100644 --- a/src/content/questions/comp2804/2015-winter-final/4/solution.md +++ b/src/content/questions/comp2804/2015-winter-final/4/solution.md @@ -1,7 +1,13 @@ $ {(2x-7y)}^{15} $ -$= \sum\_{k=4}^{15} \binom{15}{k} {(2x)}^{k} {(-7y)}^{15-k} $ +$ = \sum_{k=0}^{15} \binom{15}{k} {(2x)}^{n-k} {(-7y)}^{k} $ -$ = \binom{15}{4} 2^{4} {(-7)}^{11} x^4 y^{11}$ +We only consider $k=11$, as it results in $y^{11}$. -$ = - \binom{15}{4} 2^{4} {(7)}^{11} $ +$ = \binom{15}{11} \cdot {(2x)}^{15-11} \cdot {(-7y)}^{11} $ + +$ = \binom{15}{11} \cdot 2^{4} \cdot {(-7)}^{11} \cdot x^4 \cdot y^{11} $ + +$ = - \binom{15}{4} \cdot 2^{4} \cdot 7^{11} \cdot x^4 \cdot y^{11} $ + +Thus, the coefficient of $ x^{4}y^{11} $ in the expansion of $ {(2x-7y)}^{15} $ is $ - \binom{15}{11} \cdot {2}^{4} \cdot {7}^{11} $ \ No newline at end of file diff --git a/src/content/questions/comp2804/2016-fall-midterm/9/solution.md b/src/content/questions/comp2804/2016-fall-midterm/9/solution.md index 4707b051..80b2dcce 100644 --- a/src/content/questions/comp2804/2016-fall-midterm/9/solution.md +++ b/src/content/questions/comp2804/2016-fall-midterm/9/solution.md @@ -1,7 +1,11 @@ -$ = \sum^{50}\_{k=0} \binom{50}{k} {(5x)}^{50-k} {(-7y)}^k $ +$ = \sum^{50}_{k=0} \binom{50}{k} {(5x)}^{n-k} {(-7y)}^k $ -$ = \sum^{50}\_{k=0} \binom{50}{26} {(5x)}^{50-26} {(-7y)}^k $ +We only consider $k=26$, as it results in $y^{26}$. -$ = \binom{50}{24} {(5)}^{24} x^{24} {(-7)}^{26} y^{26} $ +$ = \binom{50}{26} \cdot {(5x)}^{50-26} \cdot {(-7y)}^{26} $ -$ = \binom{50}{26} {(5)}^{24} {(-7)}^{26} x^{24} y^{26} $ +$ = \binom{50}{26} \cdot {(5)}^{24} \cdot x^{24} \cdot {(-7)}^{26} \cdot y^{26} $ + +$ = \binom{50}{26} \cdot 5^{24} \cdot 7^{26} \cdot x^{24} \cdot y^{26} $ + +Thus, the coefficient is $ \binom{50}{26} \cdot 5^{24} \cdot 7^{26} $ \ No newline at end of file diff --git a/src/content/questions/comp2804/2017-fall-midterm/9/solution.md b/src/content/questions/comp2804/2017-fall-midterm/9/solution.md index b18e9c50..cc40c4a9 100644 --- a/src/content/questions/comp2804/2017-fall-midterm/9/solution.md +++ b/src/content/questions/comp2804/2017-fall-midterm/9/solution.md @@ -1,11 +1,11 @@ -$ = \sum^{100}\_{k=0} \binom{100}{k} {(7x)}^{k} {(-13y)}^{100-k} $ +$ = \sum^{100}_{k=0} \binom{100}{k} {(7x)}^{n-k} {(-13y)}^{k} $ -$ = \sum^{100}\_{k=0} \binom{100}{20} {(7x)}^{20} {(-13y)}^{100-20} $ +We only consider $k=80$, as it results in $y^{80}$. -$ = \binom{100}{20} {(7)}^{20} x^{20} {(-13)}^{80} y^{80} $ +$ = \binom{100}{80} \cdot {(7x)}^{100-80} \cdot {(-13y)}^{80} $ -$ = \binom{100}{20} {(7)}^{20} {(-13)}^{80} x^{20} y^{80} $ +$ = \binom{100}{80} \cdot {(7)}^{20} \cdot x^{20} \cdot {(-13)}^{80} \cdot y^{80} $ -$ = \binom{100}{20} {(7)}^{20} {(13)}^{80} x^{20} y^{80} $ +$ = \binom{100}{80} \cdot 7^{20} \cdot 13^{80} \cdot x^{20} \cdot y^{80} $ -Thus, the coefficient is $ \binom{100}{20} {(7)}^{20} {(13)}^{80} $ +Thus, the coefficient is $ \binom{100}{80} \cdot 7^{20} \cdot 13^{80} $ diff --git a/src/content/questions/comp2804/2018-fall-final/6/solution.md b/src/content/questions/comp2804/2018-fall-final/6/solution.md index 9e01cbe4..cbf2a2b5 100644 --- a/src/content/questions/comp2804/2018-fall-final/6/solution.md +++ b/src/content/questions/comp2804/2018-fall-final/6/solution.md @@ -1,7 +1,11 @@ -$ = \sum\_{k = 35}^{55} \binom{55}{k} {(5x)}^{k} {(-3y)}^{n-k} $ +$ = \sum_{k=0}^{55} \binom{55}{k} {(5x)}^{n-k} {(-3y)}^{k} $ -$ = \binom{55}{20} {(5x)}^{20} {(-3y)}^{35} $ +We only consider $k=35$, as it results in $y^{35}$. -$ = - \binom{55}{20} 5^{20} 3^{35} x^{20} y^{35}$ +$ = \binom{55}{35} \cdot {(5x)}^{55-35} \cdot {(-3y)}^{35} $ -The coefficient is $ - \binom{55}{20} 5^{20} 3^{35} $ +$ = \binom{55}{35} \cdot 5^{20} \cdot {(-3)}^{35} \cdot x^{20} \cdot y^{35}$ + +$ = - \binom{55}{35} \cdot 5^{20} \cdot 3^{35} \cdot x^{20} \cdot y^{35}$ + +The coefficient is $ - \binom{55}{35} \cdot 5^{20} \cdot 3^{35} $ diff --git a/src/content/questions/comp2804/2022-winter-final/6/solution.md b/src/content/questions/comp2804/2022-winter-final/6/solution.md index 77dc8c32..713b22cd 100644 --- a/src/content/questions/comp2804/2022-winter-final/6/solution.md +++ b/src/content/questions/comp2804/2022-winter-final/6/solution.md @@ -1,6 +1,8 @@ $ (2x - 3y)^{30} $ -$ = \sum\_{k=20}^{30} \binom{30}{k} \cdot (2x)^{30-k} \cdot (-3y)^{k} $ +$ = \sum_{k=0}^{30} \binom{30}{k} \cdot (2x)^{n-k} \cdot (-3y)^{k} $ + +We only consider $k=20$, as it results in $y^{20}$. $ = \binom{30}{20} \cdot (2x)^{30-20} \cdot (-3y)^{20} $ From 3b6a0b383dd73df2a118fc4762ed4dd39d657027 Mon Sep 17 00:00:00 2001 From: aidan <144724440+aidanln@users.noreply.github.com> Date: Mon, 16 Dec 2024 22:42:13 -0500 Subject: [PATCH 3/5] fixed an additional tag --- src/content/questions/comp2804/2015-fall-midterm/8/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/content/questions/comp2804/2015-fall-midterm/8/index.md b/src/content/questions/comp2804/2015-fall-midterm/8/index.md index 1eaad362..cdd81da6 100644 --- a/src/content/questions/comp2804/2015-fall-midterm/8/index.md +++ b/src/content/questions/comp2804/2015-fall-midterm/8/index.md @@ -8,5 +8,5 @@ solution: comp2804/2015-fall-midterm/8/solution.md tags: - comp2804 - comp2804-midterm - - comp2804-counting-solutions-of-linear-equations + - comp2804-newton's-binomial-theorem --- From 5b8a51434b53165a7cc96e3866f2bcdc26f17553 Mon Sep 17 00:00:00 2001 From: aidan <144724440+aidanln@users.noreply.github.com> Date: Mon, 16 Dec 2024 22:44:51 -0500 Subject: [PATCH 4/5] fixed minor logical error --- src/content/questions/comp2804/2015-fall-midterm/8/solution.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/content/questions/comp2804/2015-fall-midterm/8/solution.md b/src/content/questions/comp2804/2015-fall-midterm/8/solution.md index 6066b823..178edb9b 100644 --- a/src/content/questions/comp2804/2015-fall-midterm/8/solution.md +++ b/src/content/questions/comp2804/2015-fall-midterm/8/solution.md @@ -1,4 +1,4 @@ -$=\sum^{88}_{k=0} \binom{88}{k}{(3x)}^k {(-17y)}^{88-k}$ +$=\sum^{88}_{k=0} \binom{88}{k}{(3x)}^{88-k} {(-17y)}^{k}$ $=\binom{88}{7}{(3x)}^{81} {(-17y)}^{7}$ From 793aaa42a8734355f9fe77fdbd3729e67c1d9779 Mon Sep 17 00:00:00 2001 From: aidan <144724440+aidanln@users.noreply.github.com> Date: Mon, 16 Dec 2024 23:06:39 -0500 Subject: [PATCH 5/5] branch is now accurate --- .../evaluations/comp2804/2024-fall-final.md | 33 ------------------- 1 file changed, 33 deletions(-) delete mode 100644 src/content/evaluations/comp2804/2024-fall-final.md diff --git a/src/content/evaluations/comp2804/2024-fall-final.md b/src/content/evaluations/comp2804/2024-fall-final.md deleted file mode 100644 index c248fa6b..00000000 --- a/src/content/evaluations/comp2804/2024-fall-final.md +++ /dev/null @@ -1,33 +0,0 @@ ---- -title: 2024 Fall Final -path: comp2804/2024-fall-final -author: Patrick Morin -course: comp2804 -created_at: 2024-12-16 -questions: - - comp2804/2024-fall-final/1 - - comp2804/2024-fall-final/2 - - comp2804/2024-fall-final/3 - - comp2804/2024-fall-final/4 - - comp2804/2024-fall-final/5 - - comp2804/2024-fall-final/6 - - comp2804/2024-fall-final/7 - - comp2804/2024-fall-final/8 - - comp2804/2024-fall-final/9 - - comp2804/2024-fall-final/10 - - comp2804/2024-fall-final/11 - - comp2804/2024-fall-final/12 - - comp2804/2024-fall-final/13 - - comp2804/2024-fall-final/14 - - comp2804/2024-fall-final/15 - - comp2804/2024-fall-final/16 - - comp2804/2024-fall-final/17 - - comp2804/2024-fall-final/18 - - comp2804/2024-fall-final/19 - - comp2804/2024-fall-final/20 - - comp2804/2024-fall-final/21 - - comp2804/2024-fall-final/22 - - comp2804/2024-fall-final/23 - - comp2804/2024-fall-final/24 - - comp2804/2024-fall-final/25 ----