Fifty rupee note = 3
Hundred rupee note = 4
500 rupee note = 6
Totsl = 13
\(\mathrm{n}(\mathrm{S})=13 \mathrm{C}_{2}=78\)
\(\text {No. of ways the event to occur }=4\)
\(n(A)=4 C_{2}=6\)
\(\text {No. of ways the event not to occur }=72\)
The odds of event are a:b
\(6: 72 \Rightarrow 1: 12\)

\(\text { W.K.T }|\vec{a} \cdot \vec{b}|^{2}+|\vec{a} \times \vec{b}|^{2}=|\vec{a}|^{2} \cdot|\vec{b}|^{2}\)
\(|\vec{a} \times \vec{b}|^{2} =13^{2} \cdot 5^{2}-|\vec{a} \cdot \vec{b}|^{2} \)
\(=(169)(25)-3600=4225-3600 =625 \)
\(|\vec{a} \times \vec{b}| =25 \)

\(\text { Slope of first line }=\frac{-1}{(2 k-7)}\)
\(\text { Slope of second line }=\frac{-3 k}{9}\)
\(\text { They are perpendicular } \mathrm{m}_{1} \mathrm{~m}_{2}=-1\)
\(\frac{3 k}{9(2 k-7)} =-1 \)
\(k =-3(2 k-7) \)
\(k =-6 k+21 \)
\(7 k =21 \)
\(k=3 \)

\(\text { Given }{ }^{n} C_{4},{ }^{n} C_{5},{ }^{n} C_{6} \text { are in A.P }\)
\({ }^{2 n} \mathrm{C}_{5}={ }^{n} \mathrm{C}_{4}+{ }^{n} \mathrm{C}_{6}\)
\(\frac{2\lfloor n}{\lfloor n-5\lfloor 5}=\frac{\lfloor n}{\lfloor n -4\lfloor 4}+\frac{n}{\lfloor n -6\lfloor 6}\)
\(\frac{2}{\lfloor n-5\lfloor 5} =\frac{1}{\operatorname{\lfloor n}-4\lfloor 4}+\frac{1}{\lfloor n-6\lfloor 6}\)
\(\frac{2(n-4) 6}{(n-4)\lfloor n-5\lfloor 5.6}=\frac{5.6}{\lfloor-45.6\lfloor 4}+ \frac{(n-4)(n-5)}{\lfloor 6(n-4)(n-5) \lfloor n-6}\)
\(\Rightarrow \frac{12(n-4)}{\lfloor n-4\lfloor 6}=\frac{30}{\lfloor n-4\lfloor 6}+ \frac{(n-4)(n-5)}{\lfloor n-4 \lfloor 6}\)
\(12 n-48 =30+n^{2}-9 n+20 \)
\(n^{2}-21 n+98 =0 \)
\((n-14)(n-7) =0 \)
\(n=14(\text { or }) n =7 \)