Olympiad workbook
- Exam
- Olympiad workbook
- Subject
- IOQM
- Chapter
- Permutation and Combination
Olympiad workbook · IOQM · Permutation and Combination
For any finite non empty set X of integers, let (X) denote the...
For any finite non empty set of integers, let denote the largest element of and denote the number of elements in . If is the number of ordered pairs ( ) of finite non-empty sets of positive integers, such that
and can be written as where are positive integers less than 100 , find
Correct answer
43
Step-by-step solution
\begin{aligned} & A=\left\{a_1, a_2, a_3 \ldots \ldots a_p\right\} \\ & B=\left\{b_1, b_2, b_3 \ldots . b_q\right\} \\ & \qquad \begin{aligned} & a_p \cdot q=12 \\ & p \cdot b_q=11\end{aligned} \\ & \text { Case-A : } p=11, b_q=1 \\ & A=\left\{a_1, a_2, a_3 \ldots . a_{11}\right\}, B=\{1\} \\ & \Rightarrow \quad a_{11}=12, \quad q=1 \\ & \therefore { }^{11} C_{10}= total \ ways \\ & \text { Case-B : } p=1, b_q=11 \\ & \text { (1) } A=\{12\}, B=\{11\} \rightarrow 1 \text { way } \\ & \text { (2) } A=\{6\}, B=\left\{b_1, 11\right\} \rightarrow{ }^{10} C_1 \text { ways } \\ & \text { (3) } A=\{4\}, B=\left\{b_1, b_2, 11\right\} \rightarrow{ }^{10} C_2 \text { ways } \\ & \text { (4) } A=\{3\}, B=\left\{b_1, b_2, b_3, 11\right\} \rightarrow{ }^{10} C_3 \text { ways } \\ & \text { (5) } A=\{2\}, B=\left\{b_1, b_2, b_3, b_4, b_5, 11\right\} \rightarrow{ }^{10} C_5 \text { ways } \\ & \text { (6) } A=\{1\}, B=\left\{b_1, b_2, \ldots . b_{11}, 11\right\} \rightarrow 0 \text { ways } \\ & \therefore \text { Total ways }=11+1+10+45+120+252 \\ & =439 \\ & =100 \times 4+39\end{aligned}$