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(2.14)

if (n < 0 II k < 0 II k > n) return 0; int p = 1; for (int i = 1; i <= k; i++, n-4 p *-n; return p; 1

Congruent figures are figures that have the same size and the same shape; they are the exact duplicates of each other. Such figures can be moved on top of one another so that their corresponding parts line up exactly. For example, two circles having the same radius are congruent circles. Congruent triangles are triangles that have the same size and the same shape. If two triangles are congruent, their corresponding sides and angles must be congruent. Thus, congruent triangles ABC and ArBrCr in Fig. 3-1 have congruent corresponding sides (AB ArCr , BC BrCr , and /Cr). AC ArCr ) and congruent corresponding angles (/A /Ar, /B /Br, and /C

C(n, k) = n! k! (n-k) !

Fig. 3-1

This is a straightforward implementation of the formula: int comb(int, int);

(Read nABC nArBrCr as Triangle ABC is congruent to triangle A-prime, B-prime, C-prime. ) Note in the congruent triangles how corresponding equal parts may be located. Corresponding sides lie opposite congruent angles, and corresponding angles lie opposite congruent sides.

The left loop will not execute at all, because the logical test at the beginning of the loop structure is false. The right loop will execute once, however, because the logical test is not carried out until the end of the first pass through the loop. Moreover, if the string "True" is assigned to flag during this first pass through the right loop, then the execution will continue indefinitely, until flag is reassigned.

main0 -t for (int i = -1; i < 8; i++ > -c for (int j = -1; j <= i +l; j++) tout << " ' -CC comb <iA>; tout CC endl; 1 1 int factorial(int); // Returns C(n,k), the number of combinations of k from n: int comb(int n, int k)

PRINCIPLE 1:

if (n c 0 II k c 0 II k > n) return 0; return factorial(n)/(factorial(k)*factorial(n-k));

Notethatthe factorial0 function must be declared above the comb ( ) function because it function because it is is used by that function. But it does not need to be declared above the main ( > not used there.

If two triangles are congruent, then their corresponding parts are congruent. (Corresponding parts of congruent triangles are congruent.)

4.16 Write and test the dig i t ( ) function: int digit(int n, int k)

Fig. 3-2

This function returns the kth digit of the positive integer n. For example, if n is the integer 29,415, then the call digit (n, 0 ) would return the digit 5, and the call dig i t ( n , 2 > would return the digit 4. Note that the digits are numbered from right to left beginning with the zeroth digit.

PRINCIPLE 2:

This removes the right-most digit of n k times. This reduces n to an integer whose right-most digit is the same as the kth digit of the original integer. That digit is then obtained as the remainder from division by 10: int digit(int, int); main0 1 int n, k; tout << "Integer: "; tin >> n; do 1 tout << "Digit: "; tin >> k; tout << "The ' << k C-C "th digit of ' -CC n << ' is ' << digit(n, k) << endl; } while (k > 0);

(Side-Angle-Side, SAS) If two sides and the included angle of one triangle are congruent to the corresponding parts of another, then the triangles are congruent.

Note that a Do-Loop structure does not involve a formal index. Thus, the programmer must provide the logic for altering the value of the logical expression within the loop. Typically, an initial assignment is made before entering the loop structure. The logical expression is then altered at some point within the loop.

// Returns the kth digit of the integer n: int digit(int n, int k) for (int i = 0; i < k; i++) // remove right-most digit n /= 10; return n % 10;