1. How many eight digit even numbers are possible with the digits 7 5 4 5 7 5 0 7?

Is the correct answer 240? If not, please explain.

2. Express t(6,3) + t(6,2) as a single term from Pascal's Triangle.

*The (6,3) and (6,2) are subscripts of t.

1. number is even, so last digit must be 0 or 4

That leaves 7 other digits, with 3 5's and 3 7's.

So, we have 7!/3!3! * 2 = 280
How did you get 240?

each term in the triangle is the sum of the two terms above. So, we have

t(7,3)

Why do you multiply 7!/3!3! by 2?

because the last digit can be chosen in 2 ways: 0 or 4.

1. To find the number of eight-digit even numbers possible with the given digits, you need to consider the restrictions on the last digit being even. Since the last digit must be even, it can only be 4 or 0. So, let's check the two cases separately:

Case 1: The last digit is 4.
In this case, we have 7 options for the first digit (7 or 5), 7 options for the second digit (any of the remaining digits), 6 options for the third digit (any of the remaining digits except for the last digit used), and so on. Therefore, the total number of eight-digit even numbers with a last digit of 4 is 7 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 7! = 5,040.

Case 2: The last digit is 0.
Similarly, in this case, we have 7 options for the first digit, 7 options for the second digit, and so on. Hence, the total number of eight-digit even numbers with a last digit of 0 is 7 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 7! = 5,040.

Now, we need to add up the numbers of possible eight-digit even numbers from each case:
Total = 5,040 + 5,040 = 10,080.

Therefore, the correct answer is 10,080, not 240.

2. To express t(6,3) + t(6,2) as a single term from Pascal's Triangle, you need to understand that the binomial coefficients in Pascal's Triangle represent the combinations of choosing k items from a set of n items. In this case, n = 6 and k = 3 or 2.

Looking at the sixth row of Pascal's Triangle:
1 5 10 10 5 1

The term t(6,3) represents the fourth number in the row (numbering starts from zero), which is 10. Similarly, t(6,2) represents the third number, also equal to 10.

Therefore, t(6,3) + t(6,2) = 10 + 10 = 20. So, the expression can be simplified to 20 using the values from Pascal's Triangle.