If the value of $^{12} C_{2} +^{13} C_{3} +^{14} C_{4} + . . . +^{999} C_{989}$ is $^{1000} C_{a} + b$ , then $| a + b | =$
(where ( $a + b)$ is a single digit integer)

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Solution

$^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$

$^{12} C_{2} +^{13} C_{3} +^{14} C_{4} + . . . +^{999} C_{989}$
$=^{12} C_{1} +^{12} C_{2} +^{13} C_{3} +^{14} C_{4} + . . . +^{999} C_{989} -^{12} C_{1}$
$=^{13} C_{2} +^{13} C_{3} +^{14} C_{4} + . . . +^{999} C_{989} -^{12} C_{1}$
$=^{14} C_{3} +^{14} C_{4} + . . . +^{999} C_{989} -^{12} C_{1}$
$=^{1000} C_{989} -^{12} C_{1}$
$=^{1000} C_{989} - 12 =^{1000} C_{a} + b$
$∴ a = 989, b = - 12$
$| a + b | = | 989 - 12 | = 977$

Also, $^{1000} C_{989} =^{1000} C_{11}$
$^{1000} C_{11} - 12 =^{1000} C_{a} + b$
$∴ a = 11, b = - 12$
$| a + b | = | 11 - 12 | = | - 1 | = 1$

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