JEE Advance - Mathematics (1978 - No. 18)
where $$m$$ and $$n$$ are positive integers $$\left( {n \le m} \right),$$ show that
$$\left( {m,n + 1} \right) = \left( {m - 1,\,n + 1} \right) + {x^{m - n - 1}}\left( {m - 1,n} \right).$$
$$\left( {m,n + 1} \right) = \left( {m - 1,\,n + 1} \right) + {x^{m - n - 1}}\left( {m - 1,n} \right).$$
This question requires a mathematical proof.
The given equation represents a q-binomial coefficient.
The identity can be proven using combinatorial arguments or algebraic manipulation.
Understanding q-binomial coefficients and their properties is crucial.
The identity relates different q-binomial coefficients with a power of x as a coefficient.
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