\displaystyle \textbf{Question 1. }\text{In how many ways can }4\text{ letters be posted in }5\text{ letter boxes?}
\displaystyle \text{Answer:}
\displaystyle \text{Each letter can be posted in any one of the }5\text{ letter boxes.}
\displaystyle \therefore \text{Number of ways}=5^4=625
\\

\displaystyle \textbf{Question 2. }\text{Write the number of }5\text{ digit numbers that can be formed using digits }0,1\text{ and }2.
\displaystyle \text{Answer:}
\displaystyle \text{The first digit can be filled in }2\text{ ways.}
\displaystyle \text{Each of the remaining }4\text{ places can be filled in }3\text{ ways.}
\displaystyle \therefore \text{Number of numbers}=2\times3^4=162
\\

\displaystyle \textbf{Question 3. }\text{In how many ways }4\text{ women draw water from }4\text{ taps, if no tap remains unused?}
\displaystyle \text{Answer:}
\displaystyle \text{Since no tap remains unused, each tap is used by one woman.}
\displaystyle \therefore \text{Number of ways}=4!=24
\\

\displaystyle \textbf{Question 4. }\text{Write the total number of possible outcomes in a throw of }3\text{ dice in which}
\displaystyle \text{at least one of the dice shows an even number.}
\displaystyle \text{Answer:}
\displaystyle \text{Total outcomes}=6^3=216
\displaystyle \text{Outcomes in which no die shows an even number}=3^3=27
\displaystyle \therefore \text{Required number of outcomes}=216-27=189
\\

\displaystyle \textbf{Question 5. }\text{Write the number of arrangements of the letters of the word BANANA in which}
\displaystyle \text{two N's come together.}
\displaystyle \text{Answer:}
\displaystyle \text{Treat the two N's as one unit.}
\displaystyle \text{Now the letters are }(NN),B,A,A,A
\displaystyle \therefore \text{Number of arrangements}=\frac{5!}{3!}=20
\\

\displaystyle \textbf{Question 6. }\text{Write the number of ways in which }7\text{ men and }7\text{ women can sit on a round table}
\displaystyle \text{such that no two women sit together.}
\displaystyle \text{Answer:}
\displaystyle \text{First arrange }7\text{ men around the round table in }(7-1)!=6!\text{ ways.}
\displaystyle \text{There are }7\text{ gaps for }7\text{ women.}
\displaystyle \text{Women can be arranged in these gaps in }7!\text{ ways.}
\displaystyle \therefore \text{Required number of ways}=6!\times7!
\\

\displaystyle \textbf{Question 7. }\text{Write the number of words that can be formed out of the letters of the word COMMITTEE.}
\displaystyle \text{Answer:}
\displaystyle \text{The word COMMITTEE has }9\text{ letters, in which M, T and E occur twice each.}
\displaystyle \therefore \text{Number of words}=\frac{9!}{2!\,2!\,2!}=45360
\\

\displaystyle \textbf{Question 8. }\text{Write the number of all possible words that can be formed using the letters}
\displaystyle \text{of the word MATHEMATICS.}
\displaystyle \text{Answer:}
\displaystyle \text{The word MATHEMATICS has }11\text{ letters, in which M, A and T occur twice each.}
\displaystyle \therefore \text{Number of words}=\frac{11!}{2!\,2!\,2!}=4989600
\\

\displaystyle \textbf{Question 9. }\text{Write the number of ways in which }6\text{ men and }5\text{ women can dine at a round table}
\displaystyle \text{if no two women sit together.}
\displaystyle \text{Answer:}
\displaystyle \text{First arrange }6\text{ men around the round table in }(6-1)!=5!\text{ ways.}
\displaystyle \text{There are }6\text{ gaps, out of which }5\text{ gaps are selected for women.}
\displaystyle \therefore \text{Required number of ways}=5!\times{}^6C_5\times5!
\displaystyle =120\times6\times120=86400
\\

\displaystyle \textbf{Question 10. }\text{Write the number of ways, in which }5\text{ boys and }3\text{ girls can be seated in a row}
\displaystyle \text{so that each girl is between }2\text{ boys.}
\displaystyle \text{Answer:}
\displaystyle \text{Arrange }5\text{ boys in }5!\text{ ways.}
\displaystyle \text{There are }4\text{ gaps between the boys. Choose }3\text{ gaps for }3\text{ girls.}
\displaystyle \therefore \text{Required number of ways}=5!\times{}^4C_3\times3!
\displaystyle =120\times4\times6=2880
\\

\displaystyle \textbf{Question 11. }\text{Write the remainder obtained when }1!+2!+3!+\cdots+200!\text{ is divided by }14.
\displaystyle \text{Answer:}
\displaystyle \text{For }n\geq7,\ n!\text{ is divisible by }14.
\displaystyle \therefore 1!+2!+3!+\cdots+200!\equiv1!+2!+3!+4!+5!+6!\pmod{14}
\displaystyle =1+2+6+24+120+720=873
\displaystyle 873=14\times62+5
\displaystyle \therefore \text{Remainder}=5
\\

\displaystyle \textbf{Question 12. }\text{Write the number of numbers that can be formed using all four digits }1,2,3,4.
\displaystyle \text{Answer:}
\displaystyle \text{All four digits are distinct.}
\displaystyle \therefore \text{Number of numbers}=4!=24
\\

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