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NIMCET 2020 — Mathematics PYQ

NIMCET | Mathematics | 2020

If $(8 15) + (7 15) = (r n)$ , then the values of $n$ and $r$ are:

Where, $(r n) =^{n} C_{r}$

Choose the correct answer:

Explanation

1. Pascal's Identity Formula:

$(k N) + (k - 1 N) = (k N + 1)$

This rule states that adding two combinations with the same upper index ( $N$ ) and consecutive lower indices ( $k$ and $k - 1$ ) equals a combination where the upper index increases by $1$ and the lower index takes the larger of the two values ( $k$ ).

2. Compare with the Given Equation:

The equation given in the problem is:

$(8 15) + (7 15) = (r n)$

By mapping our problem to Pascal's Identity, we can identify:

$N = 15$
$k = 8$
$k - 1 = 7$

3. Calculate the Values of $n$ and $r$ :

Applying the formula:

$(8 15) + (7 15) = (8 15 + 1)$

$(8 15) + (7 15) = (8 16)$

Now, compare $(8 16)$ with the right-hand side of the given question, $(r n)$ :

$n = 16$
$r = 8$

Conclusion

The values of $n$ and $r$ are 16 and 8 respectively.

Therefore, the correct option is B) 16 and 8.