\(=\dfrac{20}{21}x\dfrac{21}{22}x\dfrac{22}{23}x...x\dfrac{1999}{2000}\)
\(=\dfrac{20}{2000}=\dfrac{1}{100}\)
=20/21x21/22x22/23x..............x1998/1999x1999/2000
=20x21x22x23x.....................x1998x1999/21x22x23x24x...............x1999x2000
=20/2000
1/100
A = (1- \(\dfrac{1}{21}\))\(\times\)( 1 - \(\dfrac{1}{22}\))\(\times\)(1-\(\dfrac{1}{23}\))\(\times\)...\(\times\)(1-\(\dfrac{1}{1999}\))\(\times\)(1- \(\dfrac{1}{2000}\))
A = \(\dfrac{21-1}{21}\)\(\times\)\(\dfrac{22-1}{22}\)\(\times\)\(\dfrac{22-1}{23}\)\(\times\)...\(\times\)\(\dfrac{1999-1}{1999}\)\(\times\)\(\dfrac{2000-1}{2000}\)
A = \(\dfrac{20}{21}\)\(\times\)\(\dfrac{21}{22}\)\(\times\)\(\dfrac{1998}{1999}\)\(\times\)\(\dfrac{1999}{2000}\)
A = \(\dfrac{21\times....\times1999}{21\times...\times1999}\) \(\times\) \(\dfrac{20}{2000}\)
A= \(\dfrac{1}{100}\)
\(\left(1-\dfrac{1}{21}\right)\cdot\left(1-\dfrac{1}{22}\right)\cdot\left(1-\dfrac{1}{23}\right)\cdot...\cdot\left(1-\dfrac{1}{1999}\right)\cdot\left(1-\dfrac{1}{2000}\right)\)
\(=\dfrac{20}{21}\cdot\dfrac{21}{22}\cdot\dfrac{22}{23}\cdot....\cdot\dfrac{1998}{1999}\cdot\dfrac{1999}{2000}\)
\(=\dfrac{20\cdot21\cdot22\cdot23\cdot24\cdot25...\cdot1998\cdot1999}{21\cdot22\cdot23\cdot24\cdot25\cdot...\cdot1999\cdot2000}\)
\(=\dfrac{20}{2000}=\dfrac{1}{100}\)
