First slide

Permutations

Question

The exponent of 3 in 100! is

Moderate

A

47
B

48
C

49
D

50

Solution

Now,
$\begin{array}{l} 100! = 1 \cdot 2 \cdot 3 \cdot \dots 98 \cdot 99 \cdot 100 \\ = (1 \cdot 2 \cdot 4 \cdot 5 \dots 98 \cdot 100) (3 \cdot 6 \cdot 9 \cdot . . 96 \cdot 99) \\ = K \cdot 3^{33} (1 \cdot 2 \cdot 3 \dots 32 \cdot 33) \\ [∵ let K = 1 \cdot 2 \cdot 4 \cdot 5 \dots 98 \cdot 100] \\ = [K (1 \cdot 2 \cdot 4 \dots 31 \cdot 32)] 3^{33} \cdot (3 \cdot 9 \cdot 12 \dots 30 \cdot 33) \\ = K_{1} \cdot 3^{33} \cdot 3^{11} (1 \cdot 2 \cdot 3 \dots 10 \cdot 11) \\ [∵ let K (1 \cdot 2 \cdot 4 \dots 31 \cdot 32) = K_{1}] \\ = K_{1} (1 \cdot 2 \cdot 4 \dots 10 \cdot 11) 3^{33} \cdot 3^{11} (3 \cdot 6 \cdot 9 \cdot 12) \\ = K_{2} 3^{33} \cdot 3^{11} \cdot 3^{4} (1 \cdot 2 \cdot 3 \cdot 4) \\ = K_{3} \cdot 3^{33} \cdot 3^{11} \cdot 3^{4} \cdot 3 [∵ let K_{2} (1 \cdot 2 \cdot 3 \cdot 4) = K_{3}] \\ = K_{3} \cdot 3^{49} \end{array}$
Hence, exponent of 3 is 49

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App