Can you explain how the 5th row of the Pascal's triangle gives the value of 11^5?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The Pascal's triangle has rows starting with one element and increasing by 2 with every subsequent row. The element in the first row is a 1. From the second row, the elements are derived by adding the elements of the earlier row that touch the element. For elements that lie in the ends one of the elements is taken as 0.

This gives the second row as 1,1; the third row is 1,2,1; the fourth row is 1,3,3,1; the fourth row is 1, 4, 6, 4, 1 and the fifth row is 1, 5, 10, 10, 5, 1.

The elements of the 5 row are:

1 = 0 + 1

5 = 1 + 4

10 = 4 + 6

10 = 4 + 6

5 = 1 + 4

1 = 1 + 0

One of the properties of the numbers in each row of the Pascal's triangle is that the number is a power of 11. 1 = 11^0, 11 = 11^1, 121 = 11^3, 14641 = 11^4.

11^5  = 161051 is derived from the row by carrying over a number if the element has more than 1 digit and adding it to the number that lies to the element on the left. Starting with 1, 5, 10, 10, 5, 1 the number obtained is (1)(5+1)(0+1)(0)(5)(1) = 161051.

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