Question 18 and 19, Exercise 6.2
Solutions of Question 18 and 19 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.
Question 18
Howmany odd numbers less than $10,000$ can be formed using the digits $0,2,3,5,6$ without repeating the digits.
Solution.
We must make $4$ digit numbers to keep the number less that $10000$
and digit at unit place must be either $3$ or $5$ to make number odd.
Possible numbers starting with $0$ and ending with $3={ }^{3} P=6$
Possible numbers starting with $0$ and ending with $5=3 p_{6}=6$
Total possible odd numbers less than $10000$ using all $5$ digits$=6+6=12$
Odd number ending at $3$ using $4$ digits out of given $5$ digits $={ }^{4} P_{3}=2$
$4$ digit odd number end at $5$ using $4$ digits out of$5={ }^{4} P_{3}=24$
$3$ digit odd number end at $3$ using $3$ digits out of $5=$ ${ }^{4} P_{2}=12$
$3$ digit odd number end at $5$ using $3$ digits out of $5=$ ${ }^{4} P_{2}=12$
$2$ digit odd number end at $3$ using $2$ digits out of $5={ }^{4} P_{1}=4$
$2$ digit odd number end at $5$ using $2$ digits out of $5={ }^{4} P_{1}=4$
Single digit odd numbers are $3$ and $5$ i.e., $2$
So total possible odd numbers less than $10000$ are $=6+6+24+12+12+4+4+2=94$
Question 19
The cheif secretary of Sindh calls a meeting of $10$ secretaries. In how many ways they be seated ar a round table if three perticular secretaries wants to sit togather?
Solution.
$3$ secretaries can sit together in $3!=6$ different ways.
Now we treat $3$ secretaries a single element and
calculate its arrangement with remaining $7$ secretaries.
So possible arrangements of $8$ elements incircle are
$$7!=5040$$
while total possible sitting arrangements are $=5040 \times 6=30240$
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