First slide

Combinations

Question

Three digit numbers xyz are formed with digits 0, 1, 2, ….., 9 such that $x \leq y \leq z$ then number of such numbers is

Difficult

A

150
B

504
C

165
D

425

Solution

x < y < z $\Rightarrow$ number of numbers = $C (9, 3)$
x = y < z $\Rightarrow$ number of numbers = $C (9, 2)$
x < y = z $\Rightarrow$ number of numbers = $C (9, 2)$
x = y = z $\Rightarrow$ number of numbers = $C (9, 1)$
Total numbers = $^{9} C_{3} +^{9} C_{2} +^{9} C_{2} + 9 = 165$

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