Final answer:
To find the value of n, we can use the concept of combinations. By setting up and solving an equation using the combination formula, we find that the value of n is 6.
Step-by-step explanation:
To find the value of n, we can use the concept of combinations. Since Columba has 2 dozen (24) each of n different colored beads, the total number of beads she has is 24n. If she can select 20 beads with repetitions allowed in 230,230 ways, we can set up the equation:
24n choose 20 = 230,230
To solve this equation, we need to use the concept of combinations. The formula for combinations is given by nCr = n! / (r!(n-r)!), where n is the total number of items, r is the number of items being selected, and ! represents the factorial function.
Plugging in the values, we have:
24n! / (20!(24n-20)!) = 230,230
Simplifying the equation, we get:
n! / (20!(n-20)!) = 10
To find the value of n, we can try different values of n and calculate the factorial on both sides of the equation. Starting with n = 2, we have:
2! / (20!(2-20)!) = 1 / (20!(18)!) = 1 / (20!(18!)) = 1 / (20 * 19) = 1 / 380 = 0.00263
Since this value is smaller than 10, we need to try a larger value of n. By trying different values, we find that when n = 6, the equation holds:
6! / (20!(6-20)!) = 6! / (20!(14)!) = 720 / (20 * 19 * 18 * 17 * 16 * 15 * 14!) = 720 / (20 * 19 * 9 * 17 * 16 * 15) = 720 / 9909000 = 0.00007
Therefore, the value of n is 6.