The number of straight lines that can be drawn in a plane with 23 given points, assuming that no three of them are collinear is

Solution
When two points are joined. a straight line is formed.
∴ Number of selections=\(^{23}C_{2}\)
\(=\frac{23\times 22}{1\times 2}= 253\)
for the first 2 days, 12 hours a day for the next 3 days but did not work on the sixth day. Then on the average how much did he work in the first six days ?

Solution
Required average=\(\left ( \frac{(2 \times 14 +3\times 12}{6} \right )\: hours\)
=^{70}⁄_{6}= 11 hours 40 minutes
The marked price of televisions is Rs. 24,000. A retailer bought it after getting successive discounts of 20 per cent and 10 per cent respectively.Then the retailer bought it at

Solution
Single equivalent discount for 20% and 10%.
\(=\left ( 20+10\frac{20\times 10}{100} \right )= 28\)%
∴ C.P. for trader = (100  28)% of 24000
\(=\frac{24000\times 72}{100}= Rs. 17280\)
50 workers can complete a job in 6 days working 8 hours a day. If 40 workers are employed to complete the job in 20 days then the number of hours they should be working per day is

Solution
⇒ 40 × 20 × x = 50 × 6 × 8
⇒\(x=\frac{50\times 6\times 8}{40\times 20}= 3\: hours\)
A invested Rs. 10,000 for 9 months and B invested Rs. 18,000 for some times in a business. If the profits of A B are equal, then the period of time for which B’s capital was invested is

Solution
A's equivalent capital for
1 month
= 10000 × 9 = Rs. 90000
∴ Period of B's investment
\(=\frac{90000}{18000 }=5 \: months\)
The Indian cricket team is to be selected out of fifteen players, five of them are bowlers. In how many ways the team can be selected so that the team contains at least three bolwers?

Solution
Ways of selections;
(i) 3 bowlers, 8 other players
(11)4 bowlers, 7 other players
(iii)5 bowlers, 6 other players
∴ Number of selections\(=^{5}C_{3}\times 10_{C_{8}}+5_{C_{4}}\times ^{10}C_{7}+^{5}C_{5}\times ^{10}C_{6}\)
\(=\frac{5\times 4\times 3}{1\times 2\times 3}\times \frac{10\times 9}{1\times 2}+5\times \frac{10\times 9\times 8}{1\times 2\times 3}+1\times \frac{10\times 9\times 8\times 7}{1\times 2\times 3\times 4}\: \: \: \: \left [ ^{n}C_{r}=^{n}C_{nr} \right ]\)
= 450 + 600 + 210 = 1260
A man swim at 5 km per hour velocity in still water. He takes 75 minutes to swim from position A to the position B and back in a river when it is flowing at 1 km per h our.The distance between A and B is

Solution
Rate downstream = 5 + 1 =6kmph
Rate upstream = 5  1 =4kmph
If AB = x km, then^{x}⁄_{6}+^{x}⁄_{4}=^{75}⁄_{60}
\(\Rightarrow \frac{2x+3x}{12 }=\frac{5}{4}\)
⇒^{x}⁄_{12}=^{5}⁄_{4}
⇒ x=3km
In a class party arranged for 43 students, 26 liked both icecream and cold drinks, 7 disliked icecream and 4 disliked both. Then the number ofstudents who liked icecream is

Solution
Required number of students
= 437  4 = 32
The volumes of three kinds of materials are in the ratio 3 : 4 : 7 and the weights of equal volumes of the three materials are in the ratio 5 : 2 : 6. If they are mixed to form a material of 65 kg then the weight of the 2nd material in the mixture is

Solution
Required ratio
=3×5:4×2:6×7
= 15: 8: 42
The two sequences 1. 4, 16,64, …. and 5, 20, 80, 320, …. are used to form a new sequence as follows :1,5,4,20,16,80,64,320, ….Then the number immediately preceding the number 1048576 is the new sequence is

Solution
S_{1}= 1. 4, 16,64,256, 1024,4096,16384,65536,262144,1048576
S_{2}=5,20,80,320,1280,5120,20480, 81920, 327680,1310720