Question Number 152817 by rexford last updated on 01/Sep/21
$$ \\ $$$${Find}\:{the}\:{coefficient}\:{of} \\ $$$$\mathrm{1}.{x}^{\mathrm{20}\:} {in}\:\left(\mathrm{1}−{x}+{x}^{\mathrm{2}} \right)^{\mathrm{20}} \\ $$$$ \\ $$$$\mathrm{2}.{x}^{\mathrm{4}} \:{in}\:\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} \right)_{} ^{\mathrm{11}} \\ $$$${please},{help}\:{me} \\ $$
Answered by puissant last updated on 01/Sep/21
$$\left.\mathrm{2}\right) \\ $$$$\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} \right)^{\mathrm{11}} =\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{11}} \left(\mathrm{1}+{x}\right)^{\mathrm{11}} \\ $$$$=\left(\mathrm{1}+\mathrm{11}{x}^{\mathrm{2}} +{C}_{\mathrm{11}} ^{\mathrm{2}} {x}^{\mathrm{4}} +…+{x}^{\mathrm{22}} \right)\left(\mathrm{1}+\mathrm{11}{x}+{C}_{\mathrm{11}} ^{\mathrm{2}} {x}^{\mathrm{2}} +…+{x}^{\mathrm{11}} \right) \\ $$$${Hence},\:{coefficient}\:{of}\:{x}^{\mathrm{4}} \:{will}\:{be}\: \\ $$$$\mathrm{1}×{C}_{\mathrm{11}} ^{\mathrm{4}} +\mathrm{11}×{C}_{\mathrm{11}} ^{\mathrm{2}} +\mathrm{1}×{C}_{\mathrm{11}} ^{\mathrm{2}} \\ $$$$={C}_{\mathrm{11}} ^{\mathrm{4}} +\mathrm{12}{C}_{\mathrm{11}} ^{\mathrm{2}} \\ $$$$=\mathrm{330}+\mathrm{660} \\ $$$$=\mathrm{990}.. \\ $$
Answered by mr W last updated on 02/Sep/21
![(1) (1−x+x^2 )^(20) =Σ_(a+b+c=20) (((20)),((a,b,c)) )1^a (−x)^b (x^2 )^c =Σ_(a+b+c=20) (((20)),((a,b,c)) )(−1)^b x^(b+2c) b+2c=20 & a+b+c=20 ⇒c=10,b=0,a=10⇒((20!)/(10!0!10!)) ⇒c=9,b=2,a=9⇒((20!)/(9!2!9!)) ⇒c=8,b=4,a=8⇒((20!)/(8!4!8!)) ⇒c=7,b=6,a=7⇒((20!)/(7!6!7!)) ⇒c=6,b=8,a=6⇒((20!)/(6!8!6!)) ... ⇒c=2,b=16,a=2⇒((20!)/(2!16!2!)) ⇒c=1,b=18,a=1⇒((20!)/(1!18!1!)) ⇒c=0,b=20,a=0⇒((20!)/(0!20!0!)) coef. of x^(20) is ((20!)/((10!)^2 0!))+((20!)/((9!)^2 2!))+((20!)/((8!)^2 4!))+...+((20!)/((1!)^2 18!))+((20!)/((0!)^2 20!)) =184 756+9 237 808+62 355 150+ +133 024 320+116 396 280+46 558 512+ +8 817 900+775 200+29 070+380+1 =377 379 369 or (1−x+x^2 )^(20) =(((1+x^3 )^(20) )/((1+x)^(20) ))=Σ_(r=0) ^(20) C_r ^(20) x^(3r) Σ_(k=0) ^∞ C_(19) ^(k+19) (−1)^k x^k 3r+k=20: r=0,k=20 ⇒C_0 ^(20) C_(19) ^(39) r=1,k=17 ⇒−C_1 ^(20) C_(19) ^(36) r=2,k=14 ⇒C_2 ^(20) C_(19) ^(33) r=3,k=11 ⇒−C_3 ^(20) C_(19) ^(30) ... r=6,k=2 ⇒C_6 ^(20) C_(19) ^(21) coef. of x^(20) is C_0 ^(20) C_(19) ^(39) −C_1 ^(20) C_(19) ^(36) +C_2 ^(20) C_(19) ^(33) −C_3 ^(20) C_(19) ^(30) +...+C_6 ^(20) C_(19) ^(21) =377 379 369 or (1−x+x^2 )^(20) =Σ_(r=0) ^(20) C_r ^(20) x^r (x−1)^r =Σ_(r=0) ^(20) [C_r ^(20) x^r Σ_(k=0) ^r C_k ^r x^(r−k) (−1)^k ] =Σ_(r=0) ^(20) Σ_(k=0) ^r (−1)^k C_r ^(20) C_k ^r x^(2r−k) 2r−k=20: k=0,r=10 ⇒C_(10) ^(20) C_0 ^(10) =((20!)/((10!)^2 0!)) k=2,r=11 ⇒C_(11) ^(20) C_2 ^(11) =((20!)/((9!)^2 2!)) k=4,r=12 ⇒C_(12) ^(20) C_4 ^(12) =((20!)/((8!)^2 4!)) ... k=20,r=20 ⇒C_(20) ^(20) C_(20) ^(20) =((20!)/((0!)^2 20!))](https://www.tinkutara.com/question/Q152819.png)
$$\left(\mathrm{1}\right) \\ $$$$\left(\mathrm{1}−{x}+{x}^{\mathrm{2}} \right)^{\mathrm{20}} =\underset{{a}+{b}+{c}=\mathrm{20}} {\sum}\begin{pmatrix}{\mathrm{20}}\\{{a},{b},{c}}\end{pmatrix}\mathrm{1}^{{a}} \left(−{x}\right)^{{b}} \left({x}^{\mathrm{2}} \right)^{{c}} \\ $$$$=\underset{{a}+{b}+{c}=\mathrm{20}} {\sum}\begin{pmatrix}{\mathrm{20}}\\{{a},{b},{c}}\end{pmatrix}\left(−\mathrm{1}\right)^{{b}} {x}^{{b}+\mathrm{2}{c}} \\ $$$${b}+\mathrm{2}{c}=\mathrm{20}\:\&\:{a}+{b}+{c}=\mathrm{20} \\ $$$$\Rightarrow{c}=\mathrm{10},{b}=\mathrm{0},{a}=\mathrm{10}\Rightarrow\frac{\mathrm{20}!}{\mathrm{10}!\mathrm{0}!\mathrm{10}!} \\ $$$$\Rightarrow{c}=\mathrm{9},{b}=\mathrm{2},{a}=\mathrm{9}\Rightarrow\frac{\mathrm{20}!}{\mathrm{9}!\mathrm{2}!\mathrm{9}!} \\ $$$$\Rightarrow{c}=\mathrm{8},{b}=\mathrm{4},{a}=\mathrm{8}\Rightarrow\frac{\mathrm{20}!}{\mathrm{8}!\mathrm{4}!\mathrm{8}!} \\ $$$$\Rightarrow{c}=\mathrm{7},{b}=\mathrm{6},{a}=\mathrm{7}\Rightarrow\frac{\mathrm{20}!}{\mathrm{7}!\mathrm{6}!\mathrm{7}!} \\ $$$$\Rightarrow{c}=\mathrm{6},{b}=\mathrm{8},{a}=\mathrm{6}\Rightarrow\frac{\mathrm{20}!}{\mathrm{6}!\mathrm{8}!\mathrm{6}!} \\ $$$$… \\ $$$$\Rightarrow{c}=\mathrm{2},{b}=\mathrm{16},{a}=\mathrm{2}\Rightarrow\frac{\mathrm{20}!}{\mathrm{2}!\mathrm{16}!\mathrm{2}!} \\ $$$$\Rightarrow{c}=\mathrm{1},{b}=\mathrm{18},{a}=\mathrm{1}\Rightarrow\frac{\mathrm{20}!}{\mathrm{1}!\mathrm{18}!\mathrm{1}!} \\ $$$$\Rightarrow{c}=\mathrm{0},{b}=\mathrm{20},{a}=\mathrm{0}\Rightarrow\frac{\mathrm{20}!}{\mathrm{0}!\mathrm{20}!\mathrm{0}!} \\ $$$${coef}.\:{of}\:{x}^{\mathrm{20}} \:{is} \\ $$$$\frac{\mathrm{20}!}{\left(\mathrm{10}!\right)^{\mathrm{2}} \mathrm{0}!}+\frac{\mathrm{20}!}{\left(\mathrm{9}!\right)^{\mathrm{2}} \mathrm{2}!}+\frac{\mathrm{20}!}{\left(\mathrm{8}!\right)^{\mathrm{2}} \mathrm{4}!}+…+\frac{\mathrm{20}!}{\left(\mathrm{1}!\right)^{\mathrm{2}} \mathrm{18}!}+\frac{\mathrm{20}!}{\left(\mathrm{0}!\right)^{\mathrm{2}} \mathrm{20}!} \\ $$$$=\mathrm{184}\:\mathrm{756}+\mathrm{9}\:\mathrm{237}\:\mathrm{808}+\mathrm{62}\:\mathrm{355}\:\mathrm{150}+ \\ $$$$+\mathrm{133}\:\mathrm{024}\:\mathrm{320}+\mathrm{116}\:\mathrm{396}\:\mathrm{280}+\mathrm{46}\:\mathrm{558}\:\mathrm{512}+ \\ $$$$+\mathrm{8}\:\mathrm{817}\:\mathrm{900}+\mathrm{775}\:\mathrm{200}+\mathrm{29}\:\mathrm{070}+\mathrm{380}+\mathrm{1} \\ $$$$=\mathrm{377}\:\mathrm{379}\:\mathrm{369} \\ $$$$ \\ $$$${or} \\ $$$$\left(\mathrm{1}−{x}+{x}^{\mathrm{2}} \right)^{\mathrm{20}} =\frac{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)^{\mathrm{20}} }{\left(\mathrm{1}+{x}\right)^{\mathrm{20}} }=\underset{{r}=\mathrm{0}} {\overset{\mathrm{20}} {\sum}}{C}_{{r}} ^{\mathrm{20}} {x}^{\mathrm{3}{r}} \underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}{C}_{\mathrm{19}} ^{{k}+\mathrm{19}} \left(−\mathrm{1}\right)^{{k}} {x}^{{k}} \\ $$$$\mathrm{3}{r}+{k}=\mathrm{20}: \\ $$$${r}=\mathrm{0},{k}=\mathrm{20}\:\Rightarrow{C}_{\mathrm{0}} ^{\mathrm{20}} {C}_{\mathrm{19}} ^{\mathrm{39}} \\ $$$${r}=\mathrm{1},{k}=\mathrm{17}\:\Rightarrow−{C}_{\mathrm{1}} ^{\mathrm{20}} {C}_{\mathrm{19}} ^{\mathrm{36}} \\ $$$${r}=\mathrm{2},{k}=\mathrm{14}\:\Rightarrow{C}_{\mathrm{2}} ^{\mathrm{20}} {C}_{\mathrm{19}} ^{\mathrm{33}} \\ $$$${r}=\mathrm{3},{k}=\mathrm{11}\:\Rightarrow−{C}_{\mathrm{3}} ^{\mathrm{20}} {C}_{\mathrm{19}} ^{\mathrm{30}} \\ $$$$… \\ $$$${r}=\mathrm{6},{k}=\mathrm{2}\:\Rightarrow{C}_{\mathrm{6}} ^{\mathrm{20}} {C}_{\mathrm{19}} ^{\mathrm{21}} \\ $$$${coef}.\:{of}\:{x}^{\mathrm{20}} \:{is} \\ $$$${C}_{\mathrm{0}} ^{\mathrm{20}} {C}_{\mathrm{19}} ^{\mathrm{39}} −{C}_{\mathrm{1}} ^{\mathrm{20}} {C}_{\mathrm{19}} ^{\mathrm{36}} +{C}_{\mathrm{2}} ^{\mathrm{20}} {C}_{\mathrm{19}} ^{\mathrm{33}} −{C}_{\mathrm{3}} ^{\mathrm{20}} {C}_{\mathrm{19}} ^{\mathrm{30}} +…+{C}_{\mathrm{6}} ^{\mathrm{20}} {C}_{\mathrm{19}} ^{\mathrm{21}} \\ $$$$=\mathrm{377}\:\mathrm{379}\:\mathrm{369} \\ $$$$ \\ $$$${or} \\ $$$$\left(\mathrm{1}−{x}+{x}^{\mathrm{2}} \right)^{\mathrm{20}} =\underset{{r}=\mathrm{0}} {\overset{\mathrm{20}} {\sum}}{C}_{{r}} ^{\mathrm{20}} {x}^{{r}} \left({x}−\mathrm{1}\right)^{{r}} \\ $$$$=\underset{{r}=\mathrm{0}} {\overset{\mathrm{20}} {\sum}}\left[{C}_{{r}} ^{\mathrm{20}} {x}^{{r}} \underset{{k}=\mathrm{0}} {\overset{{r}} {\sum}}{C}_{{k}} ^{{r}} {x}^{{r}−{k}} \left(−\mathrm{1}\right)^{{k}} \right] \\ $$$$=\underset{{r}=\mathrm{0}} {\overset{\mathrm{20}} {\sum}}\underset{{k}=\mathrm{0}} {\overset{{r}} {\sum}}\left(−\mathrm{1}\right)^{{k}} {C}_{{r}} ^{\mathrm{20}} {C}_{{k}} ^{{r}} {x}^{\mathrm{2}{r}−{k}} \\ $$$$\mathrm{2}{r}−{k}=\mathrm{20}: \\ $$$${k}=\mathrm{0},{r}=\mathrm{10}\:\Rightarrow{C}_{\mathrm{10}} ^{\mathrm{20}} {C}_{\mathrm{0}} ^{\mathrm{10}} =\frac{\mathrm{20}!}{\left(\mathrm{10}!\right)^{\mathrm{2}} \mathrm{0}!} \\ $$$${k}=\mathrm{2},{r}=\mathrm{11}\:\Rightarrow{C}_{\mathrm{11}} ^{\mathrm{20}} {C}_{\mathrm{2}} ^{\mathrm{11}} =\frac{\mathrm{20}!}{\left(\mathrm{9}!\right)^{\mathrm{2}} \mathrm{2}!} \\ $$$${k}=\mathrm{4},{r}=\mathrm{12}\:\Rightarrow{C}_{\mathrm{12}} ^{\mathrm{20}} {C}_{\mathrm{4}} ^{\mathrm{12}} =\frac{\mathrm{20}!}{\left(\mathrm{8}!\right)^{\mathrm{2}} \mathrm{4}!} \\ $$$$… \\ $$$${k}=\mathrm{20},{r}=\mathrm{20}\:\Rightarrow{C}_{\mathrm{20}} ^{\mathrm{20}} {C}_{\mathrm{20}} ^{\mathrm{20}} =\frac{\mathrm{20}!}{\left(\mathrm{0}!\right)^{\mathrm{2}} \mathrm{20}!} \\ $$
Commented by Rankut last updated on 01/Sep/21
$$ \\ $$😱😱😱
Commented by mr W last updated on 01/Sep/21
Commented by Tawa11 last updated on 01/Sep/21
$$\mathrm{Weldone}\:\mathrm{sir}. \\ $$
Commented by peter frank last updated on 03/Sep/21
$$\mathrm{GODBLESS}\:\mathrm{you} \\ $$
Answered by mr W last updated on 01/Sep/21
$$\left(\mathrm{2}\right) \\ $$$$\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} \right)^{\mathrm{11}} =\frac{\left(\mathrm{1}−{x}^{\mathrm{4}} \right)^{\mathrm{11}} }{\left(\mathrm{1}−{x}\right)^{\mathrm{11}} } \\ $$$$=\underset{{r}=\mathrm{0}} {\overset{\mathrm{11}} {\sum}}\left(−\mathrm{1}\right)^{{r}} {C}_{{r}} ^{\mathrm{11}} {x}^{\mathrm{4}{r}} \underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}{C}_{\mathrm{10}} ^{{k}+\mathrm{10}} {x}^{{k}} \\ $$$$\mathrm{4}{r}+{k}=\mathrm{4}: \\ $$$${r}=\mathrm{0},{k}=\mathrm{4}\:\Rightarrow{C}_{\mathrm{0}} ^{\mathrm{11}} {C}_{\mathrm{10}} ^{\mathrm{14}} =\mathrm{1001} \\ $$$${r}=\mathrm{1},{k}=\mathrm{0}\:\Rightarrow−{C}_{\mathrm{1}} ^{\mathrm{11}} {C}_{\mathrm{10}} ^{\mathrm{10}} =−\mathrm{11} \\ $$$${coef}.\:{of}\:{x}^{\mathrm{4}} \:{is} \\ $$$$\mathrm{1001}−\mathrm{11}=\mathrm{990} \\ $$
Answered by mathmax by abdo last updated on 02/Sep/21
$$\left(\mathrm{1}−\mathrm{x}+\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{20}} \:=\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}\right)^{\mathrm{20}} \:=\sum_{\mathrm{n}=\mathrm{0}} ^{\mathrm{20}} \:\mathrm{C}_{\mathrm{20}} ^{\mathrm{n}} \:\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}\right)^{\mathrm{n}} \\ $$$$=\sum_{\mathrm{n}=\mathrm{0}} ^{\mathrm{20}} \:\mathrm{x}^{\mathrm{n}} \:\mathrm{C}_{\mathrm{20}} ^{\mathrm{n}} \:\left(\mathrm{x}−\mathrm{1}\right)^{\mathrm{n}} \\ $$$$=\sum_{\mathrm{n}=\mathrm{0}} ^{\mathrm{20}\:} \:\mathrm{x}^{\mathrm{n}} \:\mathrm{C}_{\mathrm{20}} ^{\mathrm{n}} \:\left(\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\mathrm{x}^{\mathrm{k}} \left(−\mathrm{1}\right)^{\mathrm{n}−\mathrm{k}} \right) \\ $$$$=\sum_{\mathrm{n}=\mathrm{0}} ^{\mathrm{20}} \left(\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\mathrm{x}^{\mathrm{k}} \left(−\mathrm{1}\right)^{\mathrm{n}−\mathrm{k}} \right)\mathrm{C}_{\mathrm{20}} ^{\mathrm{n}} \:\mathrm{x}^{\mathrm{n}} \\ $$$$=\sum_{\mathrm{n}=\mathrm{0}} ^{\mathrm{20}} \left(\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\mathrm{x}^{\mathrm{k}} \left(−\mathrm{1}\right)^{\mathrm{k}} \right)\left(−\mathrm{1}\right)^{\mathrm{n}} \:\mathrm{C}_{\mathrm{20}} ^{\mathrm{n}} \:\mathrm{x}^{\mathrm{n}} \\ $$$$=\sum_{\mathrm{n}=\mathrm{0}} ^{\mathrm{20}} \left(\mathrm{C}_{\mathrm{n}} ^{\mathrm{o}} \:−\mathrm{C}_{\mathrm{n}} ^{\mathrm{1}\:} \mathrm{x}+\mathrm{C}_{\mathrm{n}} ^{\mathrm{2}} \:\mathrm{x}^{\mathrm{2}} +….\mathrm{C}_{\mathrm{n}} ^{\mathrm{n}} \:\left(−\mathrm{1}\right)^{\mathrm{n}} \:\mathrm{x}^{\mathrm{n}} \right)\left(−\mathrm{1}\right)^{\mathrm{n}} \:\mathrm{C}_{\mathrm{20}} ^{\mathrm{n}} \:\mathrm{x}^{\mathrm{n}} \\ $$$$=\sum_{\mathrm{n}=\mathrm{0}} ^{\mathrm{20}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{0}} \left(−\mathrm{1}\right)^{\mathrm{n}} \:\mathrm{C}_{\mathrm{20}} ^{\mathrm{n}\:} \:\mathrm{x}^{\mathrm{n}} \:−\sum_{\mathrm{n}=\mathrm{0}} ^{\mathrm{20}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{1}} \left(−\mathrm{1}\right)^{\mathrm{n}} \:\mathrm{C}_{\mathrm{20}} ^{\mathrm{n}\:} \mathrm{x}^{\mathrm{n}+\mathrm{1}} \:+ \\ $$$$+\sum_{\mathrm{n}=\mathrm{0}} ^{\mathrm{20}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{2}} \:\left(−\mathrm{1}\right)^{\mathrm{n}} \:\mathrm{C}_{\mathrm{20}} ^{\mathrm{n}} \:\mathrm{x}^{\mathrm{n}+\mathrm{2}} \:+….+\sum_{\mathrm{n}=\mathrm{0}} ^{\mathrm{20}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{20}} ^{\mathrm{n}} \:\mathrm{x}^{\mathrm{2n}} \:\Rightarrow \\ $$$$\mathrm{lecoefficient}\:\mathrm{de}\:\mathrm{x}^{\mathrm{n}} \:\mathrm{in}\:\mathrm{this}\:\mathrm{expansion}\:\mathrm{is} \\ $$$$\mathrm{A}_{\mathrm{n}} =\mathrm{C}_{\mathrm{20}} ^{\mathrm{0}} \:\mathrm{C}_{\mathrm{20}} ^{\mathrm{20}} \:+\mathrm{C}_{\mathrm{19}} ^{\mathrm{1}} \:\mathrm{C}_{\mathrm{20}} ^{\mathrm{19}} +\mathrm{C}_{\mathrm{18}} ^{\mathrm{2}} \:\mathrm{C}_{\mathrm{20}} ^{\mathrm{18}} \:+….+\mathrm{C}_{\mathrm{10}} ^{\mathrm{10}} \:\mathrm{C}_{\mathrm{20}} ^{\mathrm{10}} \\ $$$$\Rightarrow\mathrm{A}_{\mathrm{n}} =\left(\mathrm{C}_{\mathrm{20}} ^{\mathrm{o}} \right)^{\mathrm{2}} \:+\left(\mathrm{C}_{\mathrm{19}} ^{\mathrm{1}} \right)^{\mathrm{2}} \:+\left(\mathrm{C}_{\mathrm{18}} ^{\mathrm{2}} \right)^{\mathrm{2}} \:+…..\left(\mathrm{C}_{\mathrm{10}} ^{\mathrm{10}} \right)^{\mathrm{2}} \\ $$