ધારો કે P એ સાત અંકોની સંખ્યાઓનો ગણ છે કે જેમાં તેમના અંકોનો સરવાળો 11 થાય છે. જો P માંની સંખ્યાઓ ફક્ત 1, 2 અને 3 અંકોનો ઉપયોગ કરીને બનેલી હોય, તો ગણ \(P\) માંના ઘટકોની સંખ્યા છે :

A 173
B 164
C 158
D 161

Verified Solution

Answer & Solution

Correct Answer

(D) 161

Step-by-step Solution

Detailed explanation

(i) number of numbers created using \(1111133=\frac{7!}{5!2!} \Rightarrow 21\) (ii) number of numbers created using \(1111223=\frac{7!}{4!2!} \Rightarrow 105\) (iii) number of numbers created using \(1112222=\frac{7!}{4!3!} \Rightarrow 35\) Total \(=161\)

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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