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# help counting

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How many distinct odd-digit numbers can be written with the digits 1,2,3, 4 and 5 if no digit may be used more than once?

Jun 14, 2021

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Digits are 1, 2, 3, 4 and 5.

⇒no. of 1-digit odd no.'s  = 3

[Repetition of digits not allowed]

No. of 2-digit odd numbers

one's place has = 3 possibilities

Then, 10's place has = 4 possibilities

⇒ no. of 2-digit odd numbers = 4 × 3  = 12

No. of 3-digit odd numbers

one's place has = 3 possibilities

10's place has = 4 possibilities

then, 100's place has = 5 - 2 = 3 possibilities

⇒ no. of 3-digit odd numbers = 3 × 4 × 3 = 36

No. of 4-digit odd numbers

one's place has = 3 possibilities

10's place has = 4 possibilities

100's place has = 3 possibilities

then, 1000's place has = 2 possibilities

⇒ no. of 4-digit odd numbers = 3 × 4 × 3 × 2 = 72

No. of 5-digit odd numbers

one's place has = 3 possibilities

10's place has = 4 possibilities

100's place has = 3 possibilities

1000's place has = 2 possibilities

then, 10000's place has = 1 possibility

⇒ no. of 5-digit odd numbers = 3 × 4 × 3 × 2 × 1 = 72

∴ No. of odd numbers = 72 + 72 + 36 + 12 + 3 = 195

Jun 14, 2021