First slide

Combinations

Question

The number of different 7 digit numbers that can be written using only the three digits 1, 2 and 3 with the condition that the digit 2 occurs twice in each number is

Moderate

A

$^{7} P_{2} 2^{2}$
B

$^{7} C_{2} 2^{5}$
C

$^{7} C_{2} 5^{5}$
D

None of these

Solution

Other than 2, remaining five places are to be filled by 1 and 3
$∴$ number of ways for five places = 2 × 2 × 2 × 2 × 2 = 2⁵
For 2, selecting 2 places out of $7 =^{7} C_{2}$
$∴$ Required number of ways $=^{7} C_{2} \cdot 2^{5}$ .

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