A double-decker bus carry (u+l) passengers, u in the upperdeck and l in the lower deck. The number of ways in which the (u+l) passengers can be distributed in the two decks, if r(≤ l) particular passengers refuse to go in the upper deck and s(≤ u) refuse to sit in the lower deck, is/are

Question

A double-decker bus carry (u+l) passengers, u in the upperdeck and l in the lower deck. The number of ways in which the (u+l) passengers can be distributed in the two decks, if r(≤ l) particular passengers refuse to go in the upper deck and s(≤ u) refuse to sit in the lower deck, is/are (A)  u-s Cl-r (B) ((u+l-r-s) !/(l-r) !(u-s) !) (C)  u-s Pl-r (D) ((u+l) !/r ! s !)

Tardigrade Admin · Accepted Answer

Number of passengers to be allotted in two decker bus =(u+l)-(r+s) Number of seats available in the upper deck =(u-s) ∴ Total number of ways =((u+l-r-s) !/(l-r) !(u-s) !)

$^{u - s} C_{l - r}$

$\frac{( u + l - r - s )!}{( l - r )! ( u - s )!}$

$^{u - s} P_{l - r}$

$\frac{( u + l )!}{r ! s !}$

Number of passengers to be allotted in two decker bus
$= (u + l) - (r + s)$
Number of seats available in the upper deck $= (u - s)$
$∴$ Total number of ways $= \frac{( u + l - r - s )!}{( l - r )! ( u - s )!}$

Q. A double-decker bus carry (u+l) passengers, u in the upperdeck and l in the lower deck. The number of ways in which the (u+l) passengers can be distributed in the two decks, if r(≤l) particular passengers refuse to go in the upper deck and s(≤u) refuse to sit in the lower deck, is/are

u−sCl−r​

(l−r)!(u−s)!(u+l−r−s)!​

u−sPl−r​

r!s!(u+l)!​

Number of passengers to be allotted in two decker bus =(u+l)−(r+s) Number of seats available in the upper deck =(u−s) ∴ Total number of ways =(l−r)!(u−s)!(u+l−r−s)!​

$^{u - s} C_{l - r}$

$\frac{( u + l - r - s )!}{( l - r )! ( u - s )!}$

$^{u - s} P_{l - r}$

$\frac{( u + l )!}{r ! s !}$

Number of passengers to be allotted in two decker bus
$= (u + l) - (r + s)$
Number of seats available in the upper deck $= (u - s)$
$∴$ Total number of ways $= \frac{( u + l - r - s )!}{( l - r )! ( u - s )!}$