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Question

The number of non-negative integral solutions of $$x_{1}+x_{2}+x_{3}+x_{4}=10$$ where not all the $$x_{i}$$ are different is


A
286
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B
282
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C
166
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D
168
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Solution

The correct option is C $$166$$
Number of solutions without any conditions = $$^{ 10+4-1 }{ C }_{4-1}=286$$
Now, all terms are different can be attained in the following ways:
(a) $$ 0, 1, 4, 5: 4! = 24$$ ways
(a) $$ 0, 1, 3, 6: 4! = 24$$ ways
(a) $$ 0, 1, 2, 7: 4! = 24$$ ways
(a) $$ 0, 2, 3, 5: 4! = 24$$ ways
(a) $$ 1, 2, 3, 4: 4! = 24$$ ways
Hence, total $$=24\times 5=120$$ ways.
Hence, answer $$=286-120=166$$
Hence, (C) is correct.

Maths

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