First slide

Combinations

Question

The number of different seven 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

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

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

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

None of these

Solution

Other than 2, remaining five places can be filled by 1 and 3 for each place. The number of ways for five places
$2 \times 2 \times 2 \times 2 \times 2 = 2^{5}$
For 2, selecting 2 places out of 7 is $^{7} C_{2} .$
Hence, for required number of ways is $^{7} C_{2} \times 2^{5}$

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