First slide

Binomial theorem for positive integral Index

Question

The digit at the unit place in the numbers
19²⁰⁰⁵ + 11^{2005 _}9²⁰⁰⁵ is

Moderate

A

2
B

1
C

0
D

8

Solution

$\begin{array}{l} (19)^{2005} + (11)^{2005} - (9)^{2000} \\ = (10 + 9)^{2005} + (10 + 1)^{2005} - (9)^{2005} \end{array}$
$\begin{array}{l} = \{9^{2005} +^{2005} C_{1} (9)^{2004} \times 10 + \dots\} + (^{2005} C_{0} +^{2005} C_{1} 10 + \dots) - (9)^{2005} \\ = (^{2005} C_{1} 9^{2004} \times 10 + multiple of 10) + (1+ multipleof 10) \end{array}$
$∴ Unit digit = 1$

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