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Question Number 155075 by mr W last updated on 24/Sep/21

Commented by mr W last updated on 24/Sep/21

$w i t h e x a c t l y n t u r n s . (1 ⩽ n ⩽ 15)$ $t h e e x a m p l e a b o v e s h o w s a r o u t e$ $w i t h 7 t u r n s .$
Commented by Tawa11 last updated on 25/Sep/21

$I will love to see sir$
Commented by Tawa11 last updated on 25/Sep/21

$What of general formular for m \times n grid .$ $What is the generalization$
Commented by qaz last updated on 25/Sep/21

$(\begin{matrix} n + m \\ m \end{matrix})$
	Answered by mr W last updated on 25/Sep/21

	$t h e r e a r e t o t a l l y 16 s t e p s i n a r o u t e,$ $8 t i m e s r i g h t w a r d s a n d$ $8 t i m e s d o w n w a r d s .$ $s u c h t h a t a r o u t e h a s e x a c t l y n t u r n s,$ $t h e s e 16 s t e p s m u s t b e a r r a n g e d i n$ $n + 1 g r o u p s . i n e a c h g r o u p t h e s t e p s$ $h a v e t h e s a m e d i r e c t i o n, i . e . e i t h e r$ $r i g h t w a r d s o r d o w n w a r d s .$ $G_{1} G_{2} G_{3} . . . G_{n + 1}$ $G_{1}, G_{3}, G_{5}, . . . g r o u p s w i t h s t e p s r i g h t w a r d s$ $G_{2}, G_{4}, G_{6}, . . . g r o u p s w i t h s t e p s d o w n w a r d s$ $o r$ $G_{1}, G_{3}, G_{5}, . . . g r o u p s w i t h s t e p s d o w n w a r d s$ $G_{2}, G_{4}, G_{6}, . . . g r o u p s w i t h s t e p s r i g h t w a r d s$ $s a y t h e n u m b e r o f s t e p s i n G_{k} i s g_{k},$ $g_{1} + g_{3} + g_{5} + . . . = 8 . . (i)$ $w e h a v e m s u c h g r o u p s, w i t h m = ⌊ \frac{n}{2} ⌋ + 1 .$ $g_{2} + g_{4} + g_{6} + . . . = 8 . . (i i)$ $w e h a v e n + 1 - m s u c h g r o u p s .$ $u s i n g s t a r s & b a r s m e t h o d w e g e t$ $f o r (i) t h e r e a r e C_{m - 1}^{7} w a y s,$ $f o r (i i) t h e r e a r e C_{n - m}^{7} w a y s .$ $t o t a l l y t h e r e a r e 2 C_{m - 1}^{7} C_{n - m}^{7} w a y s .$ $s o t h e n u m b e r o f r o u t e s w i t h e x a c t l y$ $n t u r n s i s 2 C_{⌊ \frac{n}{2} ⌋}^{7} C_{n - 1 - ⌊ \frac{n}{2} ⌋}^{7} .$ $n = 1 : 2 C_{0}^{7} C_{0}^{7} = 2 r o u t e s$ $n = 2 : 2 C_{1}^{7} C_{0}^{7} = 14 r o u t e s$ $n = 3 : 2 C_{1}^{7} C_{1}^{7} = 98 r o u t e s$ $n = 4 : 2 C_{2}^{7} C_{1}^{7} = 294 r o u t e s$ $n = 5 : 2 C_{2}^{7} C_{2}^{7} = 872 r o u t e s$ $n = 6 : 2 C_{3}^{7} C_{2}^{7} = 1470 r o u t e s$ $n = 7 : 2 C_{3}^{7} C_{3}^{7} = 2450 r o u t e s$ $n = 8 : 2 C_{4}^{7} C_{3}^{7} = 2450 r o u t e s$ $n = 9 : 2 C_{4}^{7} C_{4}^{7} = 2450 r o u t e s$ $n = 10 : 2 C_{5}^{7} C_{4}^{7} = 1470 r o u t e s$ $n = 11 : 2 C_{5}^{7} C_{5}^{7} = 882 r o u t e s$ $n = 12 : 2 C_{6}^{7} C_{5}^{7} = 294 r o u t e s$ $n = 13 : 2 C_{6}^{7} C_{6}^{7} = 98 r o u t e s$ $n = 14 : 2 C_{7}^{7} C_{6}^{7} = 14 r o u t e s$ $n = 15 : 2 C_{7}^{7} C_{7}^{7} = 2 r o u t e s$ $_____________________$ $\underset{n = 1}{\sum^{15}} a l l : 12870 r o u t e s$ $c h e c k : C_{8}^{16} = \frac{16!}{8! 8!} = 12870 ✓$
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