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

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 ?

Answered by benjo_mathlover last updated on 23/Jan/21

U = 3, N = 2, S = 3, (A = L = E = 1)

words : XX = (\begin{matrix} 3 \\ 1 \end{matrix}) = 3

XY = (\begin{matrix} 6 \\ 2 \end{matrix}) \times 2 = 30

XXX = (\begin{matrix} 2 \\ 1 \end{matrix}) = 2

XXY = 3 \times (\begin{matrix} 5 \\ 1 \end{matrix}) \times \frac{3!}{2!} = 45

XYZ = 3! \times (\begin{matrix} 6 \\ 3 \end{matrix}) = 120

XXXY = 2 \times (\begin{matrix} 4 \\ 1 \end{matrix}) \times \frac{4!}{3!} = 32

XXYY = (\begin{matrix} 3 \\ 2 \end{matrix}) \times \frac{4!}{{(2!)}^{2}} = 18

WXYZ = 4! \times (\begin{matrix} 6 \\ 4 \end{matrix}) = 360

next \dots .

Commented by mr W last updated on 28/Jan/21

t h a n k s s i r!

p l e a s e c h e c k i f t h e n u m b e r o f 4 l e t t e r

w o r d s i s c o r r e c t . i g o t 778 .

Answered by mr W last updated on 10/Feb/21

Commented by mr W last updated on 10/Feb/21

3 \times U

3 \times S

2 \times N

1 \times A

1 \times L

1 \times E

e x a m p l e : 4 l e t t e r w o r d s

X X X Y \Rightarrow 2 \times 5 \times \frac{4!}{3!} = 40

X X Y Y \Rightarrow C_{2}^{3} \times \frac{4!}{2! 2!} = 18

X X Y Z \Rightarrow C_{1}^{3} \times C_{2}^{5} \times \frac{4!}{2!} = 360

X Y Z W \Rightarrow C_{4}^{6} \times 4! = 360

\Rightarrow 4 l e t t e r w o r d s : 40 + 18 + 360 \times 2 = 778

g e n e r a l l y :

{(1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6})}^{2} (1 + x + \frac{x^{2}}{2}) {(1 + x)}^{3}

2 l e t t e r w o r d s : \frac{33}{2} \times 2! = 33

3 l e t t e r w o r d s : \frac{167}{6} \times 3! = 167

4 l e t t e r w o r d s : \frac{389}{12} \times 4! = 778

5 l e t t e r w o r d s : \frac{83}{3} \times 5! = 3320

6 l e t t e r w o r d s : \frac{1283}{72} \times 6! = 12830

7 l e t t e r w o r d s : \frac{629}{72} \times 7! = 44030

8 l e t t e r w o r d s : \frac{29}{9} \times 8! = 129920

9 l e t t e r w o r d s : \frac{31}{36} \times 9! = 312480

10 l e t t e r w o r d s : \frac{11}{72} \times 10! = 554400

11 l e t t e r w o r d s : \frac{1}{72} \times 11! = 554400

t o t a l : \dots \dots

Facebook Tweet Pin