In how many ways can a Cricket team be selected from a squad of 15 players ?
15 खिलाड़ियों के दल में से एक क्रिकेट टीम का चयन कितने प्रकार से किया जा सकता है?

Options

A. 2730
B. 1370
C. 1400
D. 1365

Answer

To solve the problem of selecting a cricket team from a squad of 15 players, we need to determine how many different combinations of players can be chosen. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a squad of 15 players, and we want to select a cricket team. The problem does not specify how many players are to be selected, so we will assume we can select any number of players from 1 to 15.

### Step 2: Use the Combination Formula
The number of ways to choose \( r \) players from \( n \) players is given by the combination formula:
\[ C(n, r) = \frac{n!}{r!(n - r)!} \]
where \( n \) is the total number of players, \( r \) is the number of players to choose, and \( ! \) denotes factorial.

### Step 3: Calculate for Each Possible Team Size
Since we can select any number of players from 1 to 15, we will calculate the number of combinations for each \( r \) from 1 to 15 and then sum them up.

\[ \text{Total Ways} = C(15, 1) + C(15, 2) + C(15, 3) + \ldots + C(15, 15) \]

### Step 4: Use the Binomial Theorem
According to the binomial theorem, the sum of combinations for all possible selections from a set is given by:
\[ \sum_{r=0}^{n} C(n, r) = 2^n \]
For our case, \( n = 15 \):
\[ \sum_{r=0}^{15} C(15, r) = 2^{15} \]
However, since we are only interested in selecting at least one player, we subtract the case where no players are selected (which is \( C(15, 0) = 1 \)):
\[ \text{Total Ways} = 2^{15} - 1 \]

### Step 5: Calculate \( 2^{15} \)
Calculating \( 2^{15} \):
\[ 2^{15} = 32768 \]
Thus,
\[ \text{Total Ways} = 32768 - 1 = 32767 \]

### Final Answer
The total number of ways to select a cricket team from a squad of 15 players is **32767**.

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