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# counting

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A committee of 4 is to be chosen from a group of students. If the number of students in the group increases by 1, the number of different committees triples. How many students are in the group?

Jul 8, 2022

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So we have this equation: $$3{ n \choose 4} = {n + 1 \choose 4}$$

Simplify to: $$3 \times { n! \over {n -4}! \times 24} = { n+1! \over {n -3}! \times 24}$$

Now, further simplify: $${n! \over (n-4)! \times 8} = {(n+1)! \over (n-3)! \times 24}$$

Now, note that $$n! = (n-4)! \times (n-3) \times (n-2) \times (n-1) \times n$$.

Applying this logic to both sides gives us $${(n -3)(n-2)(n-1)n\over 8} = {n(n +1)(n-1)(n-2)\over 24}$$

Cross multiplying gives us $${(n -3)(n-2)(n-1)24n} = {8n(n +1)(n-1)(n-2)}$$

Canceling out like terms gives us $$8(n+1) = 24(n-3)$$, meaning $$n = \color{brown}\boxed5$$

Jul 9, 2022