MHT CET · Maths · Permutation Combination

_____________ numbers greater than a million can be formed with the digits \(2,3,0,3,4,2,3\).

A 60
B 360
C 420
D 120

Verified Solution

Answer & Solution

Correct Answer

(B) 360

Step-by-step Solution

Detailed explanation

All seven-digit numbers formed with the given digits are greater than a million.
Note that the digit at millions place cannot be 0 .
\(\therefore\) It can be any one of the digits \(2,3,4\).
Case I :
Digit at millions place is '2'.
Remaining 6 digits can be arranged in \(\frac{6!}{3!}\) \(=120\) ways.
Case II :
Digit at millions place is '3'.
Remaining 6 digits can be arranged in \(\frac{6!}{2!\times 2!}\) \(=180\) ways.
Case III :
Digit at millions place is '4'.
Remaining 6 digits can be arraigned in \(\frac{6!}{3!\times 2!}\) \(=60\) ways
\(\therefore\) Required number of numbers \(=120+180+60\)
\(=360\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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