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Question: In Hyderabad, there are 5 routes to Begumpet from Kukatpally and 9 routes to Dilsukhnagar from Begum...

In Hyderabad, there are 5 routes to Begumpet from Kukatpally and 9 routes to Dilsukhnagar from Begumpet. In how many ways can a person travel from Kukatpally to Dilsukhnagar via Begumpet?
(a) 14
(b) 4
(c) 40
(d) 45

Explanation

Solution

Hint: In order to find the solution of this question, we will find the number of ways to reach Begumpet from Kukatpally and then we will find the number of ways to reach Dilsukhnagar from Begumpet and then we will combine both the situations to get our answer. Also, we need to remember the concept of combination to find the number of ways in each case.

Complete step-by-step answer:
In this question, we have been asked to find the number of ways a person can travel from Kukatpally to Dilsukhnagar via Begumpet when there are 5 routes to reach Begumpet from Kukatpally and 9 ways to reach Dilsukhnagar from Begumpet.
Now, we know that when we have to choose r out of n, we use the concept of combination, that is,
nCr=n!r!(n−r)!^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}
So, we can say that when we have to choose 1 out of 5 ways to reach Begumpet from Kukatpally then there are 5C1^{5}{{C}_{1}} ways to reach. Similarly, we can say that when we have to choose 1 out of 9 ways to reach Dilsukhnagar from Begumpet, then there are 9C1^{9}{{C}_{1}} ways to reach.
Now, we can say that the total number of ways to reach Dilsukhnagar from Kukatpally via Begumpet is the product of a number of ways to reach Begumpet from Kukatpally and the number of ways to reach Dilsukhnagar from Begumpet. Therefore, we can say,
Number of ways = 5C1× 9C1^{5}{{C}_{1}}\times {{\text{ }}^{9}}{{C}_{1}}
And we know that, nCr=n!r!(n−r)!.^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}. Therefore, we can say, the number of ways,
=5!1!(5−1)!×9!1!(9−1)!=\dfrac{5!}{1!\left( 5-1 \right)!}\times \dfrac{9!}{1!\left( 9-1 \right)!}
=5!1!4!×9!1!8!=\dfrac{5!}{1!4!}\times \dfrac{9!}{1!8!}
⇒5×9=45\Rightarrow 5\times 9=45
Therefore, we can say there are a total of 45 ways to reach Dilsukhnagar from Kukatpally via Begumpet.
Hence, option (d) is the right answer.

Note: One can think of solving this question by drawing the possible number of ways and then counting but that will be more time-consuming and then solving because we might get confused or miss any of the possible routes or count any of them twice and get a wrong answer. So, it is better to use the formula of combination to get the appropriate answer.