Please help me answer number 13 Thank you 13 In Pascals tria

Please help me answer number 13. Thank you.

13. In Pascal\'s triangle, C, i Let X, Y, and z be three consecutive numbers in the C r+1 Cr nth row of Pascal\'s Triangle, ,C Y C and Z ,,C, 1. Use the nature of such that X Pascal\'s Triangle to show that X Y Z Hint: Start by finding X Y and Y Z. n+2 +2

Solution

we have

x + y = _nC_r + _nC_r+1 = _n+1C_r+1 (1)

y + z = _nC_r+1 + _nC_r+2 = _n+1C_r+2 (2)

if we add (1) + (2), we get

x + 2y + z = _n+1C_r+1 + _n+1C_r+2   (3)

but we need x + y + z, then we perform

x+2y+z - y = _n+1C_r+1 + _n+1C_r+2 - y

remember that y= _nC_r+1

x + y + z = _n+1C_r+1 + _n+1C_r+2 - _nC_r+1

we apply the the Pascal\'s Triangle again on the first term, and we get:

x + y + z = _n+2C_r+2   - _nC_r+1