The number of ways of selecting 4 cards out of 52 cards of an ordinary pack,so that exactly 3 of them are of same denomination is----------------

To determine the number of ways to select 4 cards from a standard deck of 52 cards such that exactly 3 of them are of the same denomination, we can break down the problem into manageable steps. This involves choosing the denomination for the three cards, selecting the specific cards of that denomination, and then choosing a different denomination for the fourth card. Let's go through this step-by-step.

Step 1: Choosing the Denomination for the Three Cards

First, we need to select which denomination will have 3 cards. In a standard deck, there are 13 different denominations (Ace, 2, 3, ..., 10, Jack, Queen, King). Therefore, we have:

• 13 choices for the denomination of the three cards.

Step 2: Selecting the Specific Cards

Once we have chosen the denomination, we need to select 3 cards from the 4 available cards of that denomination. The number of ways to choose 3 cards from 4 is given by the combination formula:

C(n, r) = n! / (r!(n - r)!)

In this case, we have:

• C(4, 3) = 4

So, for each denomination chosen, there are 4 ways to select the 3 cards.

Step 3: Choosing the Denomination for the Fourth Card

Next, we need to select a different denomination for the fourth card. Since we cannot choose the same denomination as the one we already selected for the three cards, we have 12 remaining denominations to choose from:

• 12 choices for the denomination of the fourth card.

Step 4: Selecting the Fourth Card

After choosing the denomination for the fourth card, we can select any one of the 4 cards available in that denomination. Thus, there are:

• 4 choices for the specific fourth card.

Putting It All Together

Now, we can combine all these choices to find the total number of ways to select the cards:

• Choices for the denomination of the three cards: 13
• Ways to choose 3 cards from that denomination: 4
• Choices for the denomination of the fourth card: 12
• Ways to choose 1 card from that denomination: 4

The total number of ways can be calculated as follows:

Total Ways = (Choices for 3 cards) × (Ways to choose 3 cards) × (Choices for 4th card's denomination) × (Ways to choose 4th card)

Substituting the values, we get:

Total Ways = 13 × 4 × 12 × 4 = 2496

Thus, the total number of ways to select 4 cards from a standard deck such that exactly 3 of them are of the same denomination is 2496.