How-many-words-with-at-least-2-letters-can-be-formed-using-the-letters-from-TINKUTARA-

Question Number 54698 by mr W last updated on 09/Feb/19

H o w m a n y w o r d s w i t h a t l e a s t 2 l e t t e r s

c a n b e f o r m e d u s i n g t h e l e t t e r s f r o m

T I N K U T A R A ?

Answered by afachri last updated on 09/Feb/19

Commented by mr W last updated on 10/Feb/19

t h a n k y o u s i r!

p l e a s e r e c h e c k y o u r r e s u l t s .

f o r e x a m p l e w o r d s w i t h 2 l e t t e r s :

w i t h 2 s a m e l e t t e r s : 2 w a y s, i . e . T T a n d A A .

w i t h 2 d i f f e r e n t l e t t e r s : w e c a n c h o o s e

2 l e t t e r s f r o m 7 l e t t e r s (i . e . T I N K U A R),

a n d w e h a v e 7 \times 6 = 42 w a y s .

t h a t i s t o s a y t h e r e a r e 2 + 42 = 44 w a y s

t o f o r m w o r d s w i t h 2 l e t t e r s, n o t 18 w a y s .

Commented by mr W last updated on 10/Feb/19

c a s e “ w o r d s w i t h 3 {l e t t e r s}^{″} :

w o r d s w i t h 2 l e t t e r s T T : T T x, T x T, x T T \Rightarrow 3 \times 6 = 18 w a y s

w o r d s w i t h 2 l e t t e r s A A : A A x, A x A, x A A \Rightarrow 3 \times 6 = 18 w a y s

w o r d s w i t h 3 d i f f e r e n t l e t t e r s : 7 \times 6 \times 5 = 210 w a y s

\Rightarrow t o t a l l y 2 \times 18 + 210 = 246 w a y s, n o t 126 w a y s .

Commented by mr W last updated on 10/Feb/19

c a s e “ w o r d s w i t h 4 {l e t t e r s}^{″} :

w o r d s w i t h 2 l e t t e r s T T a n d 2 l e t t e r s A A :

\frac{4!}{2! 2!} = 6

w o r d s w i t h 2 l e t t e r s T T, o t h e r l e t t e r s a r e d i f f e r e n t :

C_{2}^{6} \times \frac{4!}{2!} = 180

w o r d s w i t h 2 l e t t e r s A A, o t h e r l e t t e r s a r e d i f f e r e n t :

C_{2}^{6} \times \frac{4!}{2!} = 180

w o r d s w i t h d i f f e r e n t l e t t e r s :

7 \times 6 \times 5 \times 4 = 840

\Rightarrow t o t a l l y 6 + 180 \times 2 + 840 = 1206 w a y s

Commented by afachri last updated on 10/Feb/19

aahh i did it wrong . thank to illuminate

me Mr W .

Commented by afachri last updated on 10/Feb/19

so S i r, w h y t h e 5 l e t t e r s c a s e i s n o t

c o u n t e d ? ?

Commented by mr W last updated on 10/Feb/19

i h a v e n o t c h e c k e d t h e o t h e r c a s e s .

Commented by afachri last updated on 11/Feb/19

y e s S i r

Answered by mr W last updated on 12/Feb/19

w e h a v e t o t a l l y 9 l e t t e r s :

T T

A A

I N K U R

- w o r d s w i t h 2 l e t t e r s

w i t h s a m e l e t t e r s : 2 w a y s, i . e . T T a n d A A

w i t h d i f f e r e n t l e t t e r s : P_{2}^{7} = 7 \times 6 = 42 w a y s

\Rightarrow 2 + 42 = 44 w a y s

- w o r d s w i t h 3 l e t t e r s

w i t h 2 s a m e l e t t e r s T T : C_{1}^{6} \times \frac{3!}{2!} = 18 w a y s

w i t h 2 s a m e l e t t e r s A A : C_{1}^{6} \times \frac{3!}{2!} = 18 w a y s

w i t h d i f f e r e n t l e t t e r s : P_{3}^{7} = 7 \times 6 \times 5 = 210 w a y s

\Rightarrow 2 \times 18 + 210 = 246 w a y s

- w o r d s w i t h 4 l e t t e r s

w i t h 2 T T a n d 2 A A : \frac{4!}{2! 2!} = 6 w a y s

w i t h 2 T T a n d 2 o t h e r d i f f e r e n t l e t t e r s : C_{2}^{6} \times \frac{4!}{2!} = 180 w a y s

w i t h 2 A A a n d 2 o t h e r d i f f e r e n t l e t t e r s : C_{2}^{6} \times \frac{4!}{2!} = 180 w a y s

w i t h 4 d i f f e r e n t l e t t e r s : P_{4}^{7} = 7 \times 6 \times 5 \times 4 = 840 w a y s

\Rightarrow 6 + 2 \times 180 + 840 = 1206 w a y s

- w o r d s w i t h 5 l e t t e r s

T T + A A + x : C_{1}^{5} \times \frac{5!}{2! 2!} = 150 w a y s

T T + x x x : C_{3}^{6} \times \frac{5!}{2!} = 1200 w a y s

A A + x x x : C_{3}^{6} \times \frac{5!}{2!} = 1200 w a y s

w i t h 5 d i f f e r e n t l e t t e r s : P_{5}^{7} = 2520 w a y s

\Rightarrow 150 + 2 \times 1200 + 2520 = 5070 w a y s

- w o r d s w i t h 6 l e t t e r s

T T + A A + x x : C_{2}^{5} \times \frac{6!}{2! 2!} = 1800 w a y s

T T + x x x x : C_{4}^{6} \times \frac{6!}{2!} = 5400 w a y s

A A + x x x x : C_{4}^{6} \times \frac{6!}{2!} = 5400 w a y s

w i t h 6 d i f f e r e n t l e t t e r s : P_{6}^{7} = 5040 w a y s

\Rightarrow 1800 + 2 \times 5400 + 5040 = 17640 w a y s

- w o r d s w i t h 7 l e t t e r s

T T + A A + x x x : C_{3}^{5} \times \frac{7!}{2! 2!} = 12600 w a y s

T T + x x x x x : C_{5}^{6} \times \frac{7!}{2!} = 15120 w a y s

A A + x x x x x : C_{5}^{6} \times \frac{7!}{2!} = 15120 w a y s

w i t h 7 d i f f e r e n t l e t t e r s : P_{7}^{7} = 5040 w a y s

\Rightarrow 12600 + 2 \times 15120 + 5040 = 47880 w a y s

- w o r d s w i t h 8 l e t t e r s

T T + A A + x x x x : C_{4}^{5} \times \frac{8!}{2! 2!} = 50400 w a y s

T T + x x x x x x : C_{6}^{6} \times \frac{8!}{2!} = 20160 w a y s

A A + x x x x x x : C_{6}^{6} \times \frac{8!}{2!} = 20160 w a y s

\Rightarrow 50400 + 2 \times 20160 = 90720

- w o r d s w i t h 9 l e t t e r s

\Rightarrow \frac{9!}{2! 2!} = 90720

s u m : 44 + 246 + 1206 + 5070 + 17640 + 47880 + 90720 + 90720

= 253526 w a y s

Facebook Tweet Pin