Visual SolutionPYQ 2023 · Jan 24 Shift 2Standard

Question

The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition, is

(A)

120

(B)

168

(C)

220

(D)

48

Solution Path

4-digit numbers

> 7000

: first digit

7

8

, gives

48

. All 5-digit permutations:

120

. Total

= 168

01Question Setup

1/4

How many integers greater than 7000 can be formed from the digits

\{3, 5, 6, 7, 8\}

without repetition?

\text{Total} = \,?

024-Digit Numbers

2/4

For a 4-digit number

> 7000

, the first digit must be

7

8

. Remaining 3 digits chosen from 4 in order.

2 \times 4 \times 3 \times 2 = 48

035-Digit NumbersKEY INSIGHT

3/4

Every 5-digit number formed from these digits is at least 35678, which exceeds 7000. Count all permutations.

5! = 120

04Final Answer

4/4

Add the 4-digit and 5-digit counts:

48 + 120 = 168

\boxed{48 + 120 = 168}

Concepts from this question1 concepts unlocked

★

STANDARD

Count what you don't want and subtract from total: desired = total - undesired

|A| = |U| - |A^c|

When the constraint is complex, counting the complement is often far simpler. Saves 3-4 minutes on hard P&C problems.

At-least-one problemsDerangementsRestriction problems

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