Regarding Pascal's Triangle, which of the following is true?
1) The value of the term in the 13th row, 13th position is 13
2) The value of t87,1 is equal to t87,86
3)The value of t7,8 is equal to 8
4)The sum of row 8 is equal to 2^8 -1
#1 true
Not sure what 2,3 mean
#4 sum of nth row is 2^(n-1)
To solve this question, we need to understand how Pascal's Triangle works.
Pascal’s Triangle is a triangular array of numbers in which each number is the sum of the two numbers directly above it. The first and last numbers in each row are always 1.
Now, let's examine each statement one by one:
1) The value of the term in the 13th row, 13th position is 13.
To find the value in the 13th row, 13th position, we need to use the formula for the terms in Pascal's Triangle: t(row, position) = row! / ((position!) * (row - position)!), where "!" represents factorial.
Using this formula, we can calculate the value of the term in question: t(13, 13) = 13! / ((13!) * (0!)) = 1. Therefore, this statement is not true, as the value is 1, not 13.
2) The value of t87,1 is equal to t87,86.
To verify this statement, we can calculate both values. Using the formula mentioned earlier, t(87, 1) = 87! / ((1!) * (86!)) and t(87, 86) = 87! / ((86!) * (1!)).
Simplifying, we find that both expressions are equal to 87. Therefore, this statement is true.
3) The value of t7,8 is equal to 8.
To determine this, we need to calculate t(7, 8) = 7! / ((8!) * (-1!)). However, the factorial of a negative number is not defined. Therefore, this statement is not true.
4) The sum of row 8 is equal to 2^8 - 1.
To calculate the sum of the 8th row, we need to add up all the numbers in that row: 1 + 7 + 21 + 35 + 35 + 21 + 7 + 1 = 128.
On the other hand, 2^8 - 1 = 256 - 1 = 255. Therefore, this statement is not true as the sum of row 8 is 128, not 255.
In summary, the correct statements are:
- Statement 2: The value of t87,1 is equal to t87,86. (True)
- All other statements (1, 3, and 4) are false.