For 3-digit even numbers, the unit's place can be occupied by one of the 3 digits 2, 4 or 6. The remaining 5 digits can be arranged int he remaining 2 places in ₅P² ways.
\(\therefore\) By the multiplication rule, the required number of 3-digit even number is 3 \(\times\) ⁵P₂ = 3 \(\times\) 5 \(\times\) 4 = 60.

A chord is obtained by joining any two points on a circle
\(\therefore\) Total number of chords drawn through 21 points = 21C₂
\(=\frac { 21! }{ 2!19! } =\frac { 21\times 20\times 19! }{ 2\times 19! } =210\)

Given \(\frac { 2n{ C }_{ 3 } }{ n{ C }_{ 3 } } \) = \(\frac { 11 }{ 1 } \)
\(\Rightarrow\)  2nC₃ = 11.nC₃

\(\Rightarrow\) 8n - 4 = 11n - 22 \(\Rightarrow\) 4n
\(\Rightarrow\) 22-4 = 3n
\(\Rightarrow\) 3n = 18 \(\Rightarrow\) n = 6.

Given nP₄ = 20 \(\times\)nP₂.

\(\Rightarrow\) (n-2) (n-3) = 20 \(\Rightarrow\) n²-5n+6 = 20
\(\Rightarrow\) n²-5n-14 = 0
\(\Rightarrow\) (n-7)(n+2) = 0
\(\Rightarrow\) n = 7 or -2 \(\Rightarrow\) n = 7

Clearly a person can enter the room through any one of the six doors. So, there are 6 ways of entering into the room. After entering into this room, the man can come out through any one of the remaining 5 doors. So, he can come out through a different door in 5 ways.
Hence, the number of ways in which a man can enter a room through one door and come out through a different door = 6 \(\times\) 5 = 30.