The total number of odd natural numbers of six digits that can be formed using the digits 1, 3, 5, 7 if each digit is to appear in every number at least once?

1560

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1080

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2040

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840

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Solution

The correct option is A 1560
3 alike, 3 distinct
$^{4} C_{1} \times^{3} C_{3} \times \frac{6!}{3!} = 480$
2alike, 2 alike, 2 distinct
$^{4} C_{2} \times^{2} C_{2} \times \frac{6!}{2! 2!} = 1080$
$∴$ Total = 1080 + 480 = 1560

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The total number of odd natural numbers of six digits that can be formed using the digits 1, 3, 5, 7 if each digit is to appear in every number at least once?

The correct option is A 15603 alike, 3 distinct 4C1×3C3×6!3!=480 2alike, 2 alike, 2 distinct 4C2×2C2×6!2!2!=1080 ∴ Total = 1080 + 480 = 1560

The correct option is A 1560
3 alike, 3 distinct
$^{4} C_{1} \times^{3} C_{3} \times \frac{6!}{3!} = 480$
2alike, 2 alike, 2 distinct
$^{4} C_{2} \times^{2} C_{2} \times \frac{6!}{2! 2!} = 1080$
$∴$ Total = 1080 + 480 = 1560