Ram has 𝑛n trophies that he wishes to place in his main cabinet, which has space only for two trophies. If the number of trophies is increased by 3, then the number of possible ways to arrange the trophies in the main cabinet becomes 5 times the number of ways to arrange 𝑛n trophies. How many trophies does Ram have?
Question
Ram has 𝑛n trophies that he wishes to place in his main cabinet, which has space only for two trophies. If the number of trophies is increased by 3, then the number of possible ways to arrange the trophies in the main cabinet becomes 5 times the number of ways to arrange 𝑛n trophies. How many trophies does Ram have?
Solution
This problem can be solved using the concept of combinations in mathematics.
Step 1: Understand the problem Ram has n trophies and he can only place 2 trophies in his cabinet at a time. If the number of trophies is increased by 3, the number of ways to arrange the trophies in the cabinet becomes 5 times the number of ways to arrange n trophies.
Step 2: Formulate the equation The number of ways to arrange n trophies taken 2 at a time is given by the combination formula nC2. Similarly, the number of ways to arrange n+3 trophies taken 2 at a time is given by the combination formula (n+3)C2. According to the problem, (n+3)C2 = 5 * nC2.
Step 3: Simplify the equation The combination formula is given by nC2 = n! / [(n-2)! * 2!]. Plugging this into our equation gives us (n+3)! / [(n+3-2)! * 2!] = 5 * n! / [(n-2)! * 2!].
Step 4: Solve the equation Solving this equation will give us the value of n, which is the number of trophies Ram has. The solution to this equation is n = 7.
So, Ram has 7 trophies.
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