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How-many-words-with-at-least-2-letters-can-be-formed-from-UNUSUALNESS-

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Question Number 130193 by mr W last updated on 23/Jan/21
How many words with at least 2 letters can be formed from UNUSUALNESS?
$$\mathrm{How}\:\mathrm{many}\:\mathrm{words}\:\mathrm{with}\:\mathrm{at}\:\mathrm{least}\:\mathrm{2}\:\mathrm{letters} \\ $$$$\mathrm{can}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{from}\:\boldsymbol{\mathrm{UNUSUALNESS}}? \\ $$
Answered by benjo_mathlover last updated on 23/Jan/21
U=3, N=2, S=3,(A=L=E=1) words : XX = ((3),(1) ) = 3 XY = ((6),(2) ) ×2= 30 XXX = ((2),(1) ) = 2 XXY = 3× ((5),(1) ) ×((3!)/(2!)) = 45 XYZ = 3!× ((6),(3) ) = 120 XXXY = 2× ((4),(1) ) ×((4!)/(3!)) = 32 XXYY = ((3),(2) ) ×((4!)/((2!)^2 )) = 18 WXYZ = 4!× ((6),(4) ) = 360 next....
$$\:\mathrm{U}=\mathrm{3},\:\mathrm{N}=\mathrm{2},\:\mathrm{S}=\mathrm{3},\left(\mathrm{A}=\mathrm{L}=\mathrm{E}=\mathrm{1}\right) \\ $$$$\:\mathrm{words}\::\:\mathrm{XX}\:=\:\begin{pmatrix}{\mathrm{3}}\\{\mathrm{1}}\end{pmatrix}\:=\:\mathrm{3} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{XY}\:=\:\begin{pmatrix}{\mathrm{6}}\\{\mathrm{2}}\end{pmatrix}\:×\mathrm{2}=\:\mathrm{30} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{XXX}\:=\:\begin{pmatrix}{\mathrm{2}}\\{\mathrm{1}}\end{pmatrix}\:=\:\mathrm{2} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{XXY}\:=\:\mathrm{3}×\begin{pmatrix}{\mathrm{5}}\\{\mathrm{1}}\end{pmatrix}\:×\frac{\mathrm{3}!}{\mathrm{2}!}\:=\:\mathrm{45} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{XYZ}\:=\:\mathrm{3}!×\begin{pmatrix}{\mathrm{6}}\\{\mathrm{3}}\end{pmatrix}\:=\:\mathrm{120} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{XXXY}\:=\:\mathrm{2}×\begin{pmatrix}{\mathrm{4}}\\{\mathrm{1}}\end{pmatrix}\:×\frac{\mathrm{4}!}{\mathrm{3}!}\:=\:\mathrm{32} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{XXYY}\:=\:\begin{pmatrix}{\mathrm{3}}\\{\mathrm{2}}\end{pmatrix}\:×\frac{\mathrm{4}!}{\left(\mathrm{2}!\right)^{\mathrm{2}} }\:=\:\mathrm{18}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{WXYZ}\:=\:\mathrm{4}!×\begin{pmatrix}{\mathrm{6}}\\{\mathrm{4}}\end{pmatrix}\:=\:\mathrm{360} \\ $$$$\:\mathrm{next}…. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$
Commented by mr W last updated on 28/Jan/21
thanks sir! please check if the number of 4 letter words is correct. i got 778.
$${thanks}\:{sir}! \\ $$$${please}\:{check}\:{if}\:{the}\:{number}\:{of}\:\mathrm{4}\:{letter} \\ $$$${words}\:{is}\:{correct}.\:{i}\:{got}\:\mathrm{778}. \\ $$
Answered by mr W last updated on 10/Feb/21
Commented by mr W last updated on 10/Feb/21
3× U 3× S 2× N 1× A 1× L 1× E example: 4 letter words XXXY ⇒ 2×5×((4!)/(3!))=40 XXYY ⇒ C_2 ^3 ×((4!)/(2!2!))=18 XXYZ ⇒ C_1 ^3 ×C_2 ^5 ×((4!)/(2!))=360 XYZW ⇒ C_4 ^6 ×4!=360 ⇒4 letter words: 40+18+360×2=778 generally: (1+x+(x^2 /2)+(x^3 /6))^2 (1+x+(x^2 /2))(1+x)^3 2 letter words: ((33)/2)×2!=33 3 letter words: ((167)/6)×3!=167 4 letter words: ((389)/(12))×4!=778 5 letter words: ((83)/3)×5!=3320 6 letter words: ((1283)/(72))×6!=12830 7 letter words: ((629)/(72))×7!=44030 8 letter words: ((29)/9)×8!=129920 9 letter words: ((31)/(36))×9!=312480 10 letter words: ((11)/(72))×10!=554400 11 letter words: (1/(72))×11!=554400 total: ......
$$\mathrm{3}×\:{U} \\ $$$$\mathrm{3}×\:{S} \\ $$$$\mathrm{2}×\:{N} \\ $$$$\mathrm{1}×\:{A} \\ $$$$\mathrm{1}×\:{L} \\ $$$$\mathrm{1}×\:{E} \\ $$$$ \\ $$$${example}:\:\mathrm{4}\:{letter}\:{words} \\ $$$${XXXY}\:\:\Rightarrow\:\mathrm{2}×\mathrm{5}×\frac{\mathrm{4}!}{\mathrm{3}!}=\mathrm{40} \\ $$$${XXYY}\:\Rightarrow\:{C}_{\mathrm{2}} ^{\mathrm{3}} ×\frac{\mathrm{4}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{18} \\ $$$${XXYZ}\:\Rightarrow\:{C}_{\mathrm{1}} ^{\mathrm{3}} ×{C}_{\mathrm{2}} ^{\mathrm{5}} ×\frac{\mathrm{4}!}{\mathrm{2}!}=\mathrm{360} \\ $$$${XYZW}\:\Rightarrow\:{C}_{\mathrm{4}} ^{\mathrm{6}} ×\mathrm{4}!=\mathrm{360} \\ $$$$\Rightarrow\mathrm{4}\:{letter}\:{words}:\:\mathrm{40}+\mathrm{18}+\mathrm{360}×\mathrm{2}=\mathrm{778} \\ $$$$ \\ $$$${generally}: \\ $$$$\left(\mathrm{1}+{x}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}}+\frac{{x}^{\mathrm{3}} }{\mathrm{6}}\right)^{\mathrm{2}} \left(\mathrm{1}+{x}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)\left(\mathrm{1}+{x}\right)^{\mathrm{3}} \\ $$$$\mathrm{2}\:{letter}\:{words}:\:\frac{\mathrm{33}}{\mathrm{2}}×\mathrm{2}!=\mathrm{33} \\ $$$$\mathrm{3}\:{letter}\:{words}:\:\frac{\mathrm{167}}{\mathrm{6}}×\mathrm{3}!=\mathrm{167} \\ $$$$\mathrm{4}\:{letter}\:{words}:\:\frac{\mathrm{389}}{\mathrm{12}}×\mathrm{4}!=\mathrm{778} \\ $$$$\mathrm{5}\:{letter}\:{words}:\:\frac{\mathrm{83}}{\mathrm{3}}×\mathrm{5}!=\mathrm{3320} \\ $$$$\mathrm{6}\:{letter}\:{words}:\:\frac{\mathrm{1283}}{\mathrm{72}}×\mathrm{6}!=\mathrm{12830} \\ $$$$\mathrm{7}\:{letter}\:{words}:\:\frac{\mathrm{629}}{\mathrm{72}}×\mathrm{7}!=\mathrm{44030} \\ $$$$\mathrm{8}\:{letter}\:{words}:\:\frac{\mathrm{29}}{\mathrm{9}}×\mathrm{8}!=\mathrm{129920} \\ $$$$\mathrm{9}\:{letter}\:{words}:\:\frac{\mathrm{31}}{\mathrm{36}}×\mathrm{9}!=\mathrm{312480} \\ $$$$\mathrm{10}\:{letter}\:{words}:\:\frac{\mathrm{11}}{\mathrm{72}}×\mathrm{10}!=\mathrm{554400} \\ $$$$\mathrm{11}\:{letter}\:{words}:\:\frac{\mathrm{1}}{\mathrm{72}}×\mathrm{11}!=\mathrm{554400} \\ $$$${total}:\:…… \\ $$

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