jee-advanced 2005 Q15

jee-advanced · India · screening Combinations & Selection Lattice Path Counting

15. A rectangle with sides ( $2 n - 1$ ) and ( $2 m - 1$ ) is divided into squares of unit length. The number of rectangle which can be formed with sides of odd length is :
(a) $m ^ { 2 } n ^ { 2 }$
(b) $m n ( m + 1 ) ( n + 1 )$
(c) $4 ^ { m + n - 1 }$
(d) none of these

Let $S$ be the set of all polynomials $P ( x )$ of degree less than or equal to 2 which satisfy the conditions $P ( 1 ) = 1 , P ( 0 ) = 0$ and $P ^ { \prime } ( x ) > 0$ for all $x \in [ 0,1 ]$. Then

15. A rectangle with sides ( $2 n - 1$ ) and ( $2 m - 1$ ) is divided into squares of unit length. The number of rectangle which can be formed with sides of odd length is :\\
(a) $m ^ { 2 } n ^ { 2 }$\\
(b) $m n ( m + 1 ) ( n + 1 )$\\
(c) $4 ^ { m + n - 1 }$\\
(d) none of these\\

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