Number of integral solutions to the equation x + y + z = 21 , where x ≥ 1 , y ≥ 3 , z ≥ 4 …

Number of integral solutions to the equation x+y+z=21, where x≥1,y≥3,z≥4, is equal to _____ .

Solution

We have,

x≥1, y≥3, z≥4

⇒x-1≥0, y-3≥0, z-4≥0

So,

x+y+z=21

⇒x-1+y-3+z-4=13

So, number of non-negative integral solutions is

=C3-113+3-1=C215=15×142=105

Asked in: JEE Main 2023 (01 Feb Shift 2)

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