If-only-downward-motion-along-lines-is-allowed-what-is-the-total-number-of-paths-from-point-P-to-point-Q-in-the-figure-below-

Question Number 17938 by Tinkutara last updated on 12/Jul/17

$$\mathrm{If}\:\mathrm{only}\:\mathrm{downward}\:\mathrm{motion}\:\mathrm{along}\:\mathrm{lines}\:\mathrm{is} \\ $$$$\mathrm{allowed},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{paths}\:\mathrm{from}\:\mathrm{point}\:\mathrm{P}\:\mathrm{to}\:\mathrm{point}\:\mathrm{Q}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{figure}\:\mathrm{below}? \\ $$

Commented by Tinkutara last updated on 12/Jul/17

Commented by ajfour last updated on 13/Jul/17

3 leftdown and 3 rightdown steps to be chosen. Number of ways =coeff. of x^3 y^3 in the expansion of (x+y)^6 =^6 C_3 =20.

$$\mathrm{3}\:\mathrm{leftdown}\:\mathrm{and}\:\mathrm{3}\:\mathrm{rightdown}\:\mathrm{steps} \\ $$$$\mathrm{to}\:\mathrm{be}\:\mathrm{chosen}. \\ $$$$\mathrm{Number}\:\mathrm{of}\:\mathrm{ways}\:=\mathrm{coeff}.\:\mathrm{of}\:\mathrm{x}^{\mathrm{3}} \mathrm{y}^{\mathrm{3}} \:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{x}+\mathrm{y}\right)^{\mathrm{6}} =^{\mathrm{6}} \mathrm{C}_{\mathrm{3}} =\mathrm{20}. \\ $$

Answered by alex041103 last updated on 12/Jul/17